[{"content":" $\\newcommand{\\N}{\\mathbb{N}} \\newcommand{\\Z}{\\mathbb{Z}} \\newcommand{\\Q}{\\mathbb{Q}} \\newcommand{\\R}{\\mathbb{R}} \\newcommand{\\LP}{\\left(} \\newcommand{\\RP}{\\right)} \\newcommand{\\LS}{\\left\\lbrace} \\newcommand{\\RS}{\\right\\rbrace} \\newcommand{\\LA}{\\left\\langle} \\newcommand{\\RA}{\\right\\rangle} \\newcommand{\\LB}{\\left[} \\newcommand{\\RB}{\\right]} \\newcommand{\\MM}{\\ \\middle|\\ } \\newcommand{\\exp}{\\text{exp}} \\newcommand{\\abs}[1]{\\left\\vert#1\\right\\vert} \\newcommand{\\msr}[1]{m\\left(#1\\right)} \\newcommand{\\inv}[1]{#1^{-1}} \\newcommand{\\bkt}[1]{\\LA \\img{#1}\\RA} \\require{color}$ Below is a tool designed to draw the tangle represented by a weighted planar tangle tree [1].\nNote The rendered image is an `svg`, a vector format. This means it can be scaled infinitely. If you need to view it at a larger scale or would like to postprocess the image, you can download it and use a tool such as Inkscape to edit it. Alert I have done very little testing with this tool. I'm reasonably confident that valid input generates valid output. I have no idea what happens when invalid input is fed to the tool. If you find a bug, please report it. Report an Issue! Instructions Note I've given a good number of dials here to mess with. I've set some defaults that seem to work well. In the \u0026ldquo;Tangle\u0026rdquo; field, input a linearized weighted planar tangle tree (information found in my thesis link to come). Configure the tool: In the \u0026ldquo;Strand Color\u0026rdquo; field, set the color of the strands of the tangle. For print you almost certainly want #000000. In the \u0026ldquo;Crossing Color\u0026rdquo; field, set the color of the crossing for the tangle. When an overstrand is drawn, a $2\\times$-sized copy is placed underneath it. This gives the appearance of the understrand being broken. For print you almost certainly want #ffffff. In the \u0026ldquo;Eccentricity\u0026rdquo; field, select a number from 0 to 100. This will tell the tool how far away from a path vertex to place control points. Play around until you find settings you like. In the \u0026ldquo;String Size\u0026rdquo; field, select a positive integer. This will be, in pixels, how large to make the strands of the tangle. A crossing will always have a height and width of $\\text{\u0026ldquo;String Size\u0026rdquo;}\\cdot 11$. In the \u0026ldquo;Gap Size\u0026rdquo; field, select a positive integer. This dictates how far apart units of a tangle should be placed. Play around until you find something you like. Press the \u0026ldquo;Draw\u0026rdquo; button to render the tangle as an svg. If you\u0026rsquo;re happy with the picture, you can download it by pressing the \u0026ldquo;Download\u0026rdquo; button. Tangle: Strand Color:\nCrossing Color:\nEccentricity: String Size:\nGap Size:\nSVG size (in px): Columns: Draw Bulk print tangles up to TCN 10: ($-1$ is the $\\infty$ tangle and bounds are inclusive) From: Up to: Bulk draw tangles (this could take a while) Download F. Bonahon and\u0026#32;L. Siebenmann,\u0026#32;New Geometric Splittings of Classical Knots and the Classification and Symmetries of Arborescent Knots,\u0026#32;https://dornsife.usc.edu/francis-bonahon/wp-content/uploads/sites/205/2023/06/BonSieb-compressed.pdf,\u0026#32;2016.\u0026#32; ","date":"16 August 2025","externalUrl":null,"permalink":"/resources/tools/draw_wptt/","section":"Resources","summary":"","title":"Draw Weighted Planar Tangle Trees","type":"resources"},{"content":" Context of the find An LED lamp I found at Marshalls.\nKnot Information Number of components 1\nNumber of crossings 3\nKnotinfo link $3_1$\nReport an Issue! ","date":"6 August 2025","externalUrl":null,"permalink":"/misc/knots_in_the_wild/0006/","section":"","summary":"","title":"An LED lamp I found at Marshalls.","type":"misc"},{"content":" $\\newcommand{\\N}{\\mathbb{N}} \\newcommand{\\Z}{\\mathbb{Z}} \\newcommand{\\Q}{\\mathbb{Q}} \\newcommand{\\R}{\\mathbb{R}} \\newcommand{\\LP}{\\left(} \\newcommand{\\RP}{\\right)} \\newcommand{\\LS}{\\left\\lbrace} \\newcommand{\\RS}{\\right\\rbrace} \\newcommand{\\LA}{\\left\\langle} \\newcommand{\\RA}{\\right\\rangle} \\newcommand{\\LB}{\\left[} \\newcommand{\\RB}{\\right]} \\newcommand{\\MM}{\\ \\middle|\\ } \\newcommand{\\exp}{\\text{exp}} \\newcommand{\\abs}[1]{\\left\\vert#1\\right\\vert} \\newcommand{\\msr}[1]{m\\left(#1\\right)} \\newcommand{\\inv}[1]{#1^{-1}} \\newcommand{\\bkt}[1]{\\LA \\img{#1}\\RA} \\require{color}$ Context of the find A wooden table decoration I found at Aldi.\nKnot Information Number of components 3\nNumber of crossings 8\nArborescent symbol $$\\LP[-2]\\LP[2][-2]\\RP[2]\\RP$$\nKnotinfo link $L8_{n3}$\nReport an Issue! ","date":"5 August 2025","externalUrl":null,"permalink":"/misc/knots_in_the_wild/0005/","section":"","summary":"","title":"A wooden table decoration I found at Aldi.","type":"misc"},{"content":" Context of the find A string of beads I found at Joann Fabrics.\nKnot Information I\u0026rsquo;m going to give info for each of the beads.\nNumber of components 2\nNumber of crossings 8\nKnotinfo link $L8_{a14}$\nReport an Issue! ","date":"4 August 2025","externalUrl":null,"permalink":"/misc/knots_in_the_wild/0004/","section":"","summary":"","title":"A string of beads I found at Joann Fabrics.","type":"misc"},{"content":" Context of the find A string of beads I found at Joann Fabrics.\nKnot Information I\u0026rsquo;m going to give info for each of the beads.\nNumber of components 1\nNumber of crossings 8\nKnotinfo link $8_{18}$\nReport an Issue! ","date":"3 August 2025","externalUrl":null,"permalink":"/misc/knots_in_the_wild/0003/","section":"","summary":"","title":"A string of beads I found at Joann Fabrics.","type":"misc"},{"content":" Context of the find A monkey\u0026rsquo;s paw pillow I found at Marshalls.\nKnot Information This one is hard, it\u0026rsquo;s unclear what is happening in the middle of the pillow. As it turns out, this is actually not a knot (in the mathematical sense). When you pull on the string, you find out that the ends are not connected (you can see this in the background), so this pillow is isotopic to a point.\nReport an Issue! ","date":"2 August 2025","externalUrl":null,"permalink":"/misc/knots_in_the_wild/0002/","section":"","summary":"","title":"A monkey's paw pillow I found at Marshalls.","type":"misc"},{"content":" Context of the find A string of beads I found at Joann Fabrics.\nKnot Information I\u0026rsquo;m going to give info for each of the beads.\nNumber of components 1\nNumber of crossings 7\nKnotinfo link $7_4$\nReport an Issue! ","date":"1 August 2025","externalUrl":null,"permalink":"/misc/knots_in_the_wild/0001/","section":"","summary":"","title":"A string of beads I found at Joann Fabrics.","type":"misc"},{"content":" Ann Arbor Stuff To Do And See Ann Arbor Hands-On Museum Art Fair Big House Tours Michigan and State Theater Pinball Pete\u0026rsquo;s Summer Festival The Ark The diag The Law Quadrangle U-M Museum of Natural History U-M Sports Places To Eat Downtown Blank Slate Creamery Comet Coffee Fleetwood Diner Frita Batidos Ann Arbor Good Time Charley\u0026rsquo;s Hop Cat Jerusalem Garden Jolly Pumpkin Lab Cafe Namaste Flavours Ann Arbor Neo Papolis No Thai Pizza House RoosRoast Slurping Turtle The Pretzel Bell Zingermans Bubble Tea Area Chelas Pilars The Bomber Not Ann Arbor Stuff To Do And See Art in the park Detroit Lions Detroit Pistons Detroit Red Wings Detroit Science Center Detroit Tigers Detroit Zoo DIA Ford Rouge Factory Tour Greenfield Village Guernsey Farms Dairy Mexican Town Motown Museum Old-Time Baseball Plymouth Orchard Somerset Collection The Henry Ford Third Man Records Places To Eat Aahar Chilangos Mi Pueblo Shangri-La The Red Dot The most Detroit food history Lafayette Coney Island or American Coney Island ","date":"28 May 2025","externalUrl":null,"permalink":"/resources/general/mi_guide/","section":"Resources","summary":"Brief Guide To Ann Arbor And Metro Detroit","title":"Brief Guide To Ann Arbor And Metro Detroit","type":"resources"},{"content":" $\\newcommand{\\N}{\\mathbb{N}} \\newcommand{\\Z}{\\mathbb{Z}} \\newcommand{\\Q}{\\mathbb{Q}} \\newcommand{\\R}{\\mathbb{R}} \\newcommand{\\LP}{\\left(} \\newcommand{\\RP}{\\right)} \\newcommand{\\LS}{\\left\\lbrace} \\newcommand{\\RS}{\\right\\rbrace} \\newcommand{\\LA}{\\left\\langle} \\newcommand{\\RA}{\\right\\rangle} \\newcommand{\\LB}{\\left[} \\newcommand{\\RB}{\\right]} \\newcommand{\\MM}{\\ \\middle|\\ } \\newcommand{\\exp}{\\text{exp}} \\newcommand{\\abs}[1]{\\left\\vert#1\\right\\vert} \\newcommand{\\msr}[1]{m\\left(#1\\right)} \\newcommand{\\inv}[1]{#1^{-1}} \\newcommand{\\bkt}[1]{\\LA \\img{#1}\\RA} \\require{color}$ Instructions Input your url (string) into the url field. Select your foreground and background colors. Press the \u0026ldquo;create qr code\u0026rdquo; button. You can then verify that the qr code works and download it with the download button.\nURL:\nCreate qr code Foreground color:\nBackground color:\nDownload ","date":"15 April 2025","externalUrl":null,"permalink":"/resources/tools/qr_svg/","section":"Resources","summary":"A tool","title":"QR code svg generator","type":"resources"},{"content":" $\\newcommand{\\N}{\\mathbb{N}} \\newcommand{\\Z}{\\mathbb{Z}} \\newcommand{\\Q}{\\mathbb{Q}} \\newcommand{\\R}{\\mathbb{R}} \\newcommand{\\LP}{\\left(} \\newcommand{\\RP}{\\right)} \\newcommand{\\LS}{\\left\\lbrace} \\newcommand{\\RS}{\\right\\rbrace} \\newcommand{\\LA}{\\left\\langle} \\newcommand{\\RA}{\\right\\rangle} \\newcommand{\\LB}{\\left[} \\newcommand{\\RB}{\\right]} \\newcommand{\\MM}{\\ \\middle|\\ } \\newcommand{\\exp}{\\text{exp}} \\newcommand{\\abs}[1]{\\left\\vert#1\\right\\vert} \\newcommand{\\msr}[1]{m\\left(#1\\right)} \\newcommand{\\inv}[1]{#1^{-1}} \\newcommand{\\bkt}[1]{\\LA \\img{#1}\\RA} \\require{color}$ Here\u0026rsquo;s a button to take you to the materials as pdf and zip files Click me to go to the page where you can download the materials! Here\u0026rsquo;s a link to the main page of the GitHub repository Joecstarr/MfaCoBPM The contents of this repository describe the outline for a course on basic project management. There are three document sets each described in the README. TeX 0 0 Here\u0026rsquo;s a link to the README of the GitHub repository Materials for a Course on Basic Project Management The contents of this repository describe the outline for a course on basic project management. There are three document sets each described in the following sections.\nNote to Reader If you find an issue with this repository or have a question please feel free to open an issue. I\u0026rsquo;ve included templates for the following issues:\n🐞 Spelling and Grammar 🤷 Clarity ❓ Question 🚀 Enhancement What\u0026rsquo;s in This Repository? Project plan The project plan ./documents/plan/project_plan.md is the primary planning document for the project.\nRisk Management Plan The risk management plan ./documents/risk/risk_management_plan.md is gives an overview of things that can go wrong in a project and how to handle things when they do go wrong.\nProject Schedule The project schedule ./documents/schedule/schedule.md gives a list of all tasks involved with project competition.\nCourse Design Documents The course design ./course_planning/course_design.md contains the overall course goals as well as per week outlines of the course content.\nExample Project Plan The contents of ./example are a sample example plan created for a hypothetical birdhouse building project.\n[!NOTE] This example plan is not perfect. You should think about this example as a step below a \u0026ldquo;minimum\u0026rdquo; quality for a project plan. Content written here should serve to inform the essence of what should go in each section.\nCite Me 📃 Bibtex and APA on the right sidebar of github.\nCitations Boehm, B. W. (1989). Software risk management. IEEE Computer Society Press. https://books.google.com/books?id=ittWAAAAMAAJ Col Lawrence Nixon. (n.d.). RISK MANAGEMENT (RM) GUIDELINES AND TOOLS. DEPARTMENT OF THE AIR FORCE. Garvey, P. R. (2008). Analytical Methods for Risk Management (0 ed.). Chapman and Hall/CRC. https://doi.org/10.1201/9781420011395 Internationale Elektrotechnische Kommission \u0026amp; Internationale Organisation für Normung (Eds.). (2019). Risk management: Risk assessment techniques (Edition 2.0). IEC Central Office, Commission Electrotechnique Internationale. Pressman, R. S. (2015). Software engineering: A practitioner\u0026rsquo;s approach (Eighth edition). McGraw-Hill Education. Thomsett, R. (1992). The Indiana Jones School of Risk Management. American Programmer, 5(7), 10-18. Wiggins, G. P., \u0026amp; McTighe, J. (2008). Understanding by design (Expanded 2nd ed, [Nachdr.]). Association for Supervision and Curriculum Development. Williams, R. C., Walker, J. A., \u0026amp; Dorofee, A. J. (1997). Putting risk management into practice. IEEE Software, 14(3), 75-82. https://doi.org/10.1109/52.589240 Yourdon, E. (1995). When good enough software is best. IEEE Software, 12(3), 79-81. https://doi.org/10.1109/52.382191 License ⚖️ Materials for a Course on Teaching Undergraduate Research by Joseph Starr and Lori Adams is licensed under CC BY-NC-SA 4.0\n","date":"10 February 2025","externalUrl":null,"permalink":"/resources/course_plans/mfacobpm/","section":"Resources","summary":"-| The contents of this repository describe the outline for a course on basic project management. There are three document sets each described in the README.","title":"Materials for a Course on Basic Project Management","type":"resources"},{"content":" $\\newcommand{\\N}{\\mathbb{N}} \\newcommand{\\Z}{\\mathbb{Z}} \\newcommand{\\Q}{\\mathbb{Q}} \\newcommand{\\R}{\\mathbb{R}} \\newcommand{\\LP}{\\left(} \\newcommand{\\RP}{\\right)} \\newcommand{\\LS}{\\left\\lbrace} \\newcommand{\\RS}{\\right\\rbrace} \\newcommand{\\LA}{\\left\\langle} \\newcommand{\\RA}{\\right\\rangle} \\newcommand{\\LB}{\\left[} \\newcommand{\\RB}{\\right]} \\newcommand{\\MM}{\\ \\middle|\\ } \\newcommand{\\exp}{\\text{exp}} \\newcommand{\\abs}[1]{\\left\\vert#1\\right\\vert} \\newcommand{\\msr}[1]{m\\left(#1\\right)} \\newcommand{\\inv}[1]{#1^{-1}} \\newcommand{\\bkt}[1]{\\LA \\img{#1}\\RA} \\require{color}$ Here\u0026rsquo;s a button to take you to the materials as pdf and zip files Click me to go to the page where you can download the materials! Here\u0026rsquo;s a link to the main page of the GitHub repository Joecstarr/MfaCoTUR The contents of this repository describe the outline for a course on teaching undergraduate research. TeX 0 0 Here\u0026rsquo;s a link the README of the GitHub repository Materials for a Course on Teaching Undergraduate Research The contents of this repository describe the outline for a course on teaching undergraduate research. There are twp document sets each described in the following sections.\nNote to Reader If you find an issue with this repository or have a question please feel free to open an issue. I\u0026rsquo;ve included templates for the following issues:\n🐞 Spelling and Grammar 🤷 Clarity ❓ Question 🚀 Enhancement What\u0026rsquo;s in This Repository? Course Design Documents The course design ./course_planning/course_design.md contains the overall course goals as well as per week outlines of the course content. Additionally, you can find materials used in the course design.\nModel Lesson The files for my model lesson given in week 2 of the course.\nCite Me 📃 Bibtex and APA on the right sidebar of github.\nCitations Jupyter Book Contributors, Cockett, R., Koch, F., Purves, S., Hollands, A., Yuxi Wang, … Josh Borrow. (2025). jupyter-book/mystmd: v1.3.22 (Version mystmd@1.3.22). Zenodo. https://doi.org/10.5281/ZENODO.14805610 Fisher, D., \u0026amp; Frey, N. (2013). Better Learning Through Structured Teaching: A Framework for the Gradual Release of Responsibility. ASCD. Hoffmann, D., \u0026amp; Lenoch, S. (2013). Teaching Your Research: A Workshop to Teach Curriculum Design to Graduate Students and Post-doctoral Fellows. Medical Science Educator, 23(3), 336-345. Stommel, J. (2023). Undoing the Grade: Why We Grade, and How to Stop. Hybrid Pedagogy Inc.. Wiggins, G., \u0026amp; McTighe, J. (2008). Understanding by Design. Association for Supervision and Curriculum Development. License ⚖️ Materials for a Course on Teaching Undergraduate Research by Joseph Starr and Lori Adams is licensed under CC BY-NC-SA 4.0\n","date":"10 February 2025","externalUrl":null,"permalink":"/resources/course_plans/mfacotur/","section":"Resources","summary":"-| The contents of this repository describe the outline for a course on teaching undergraduate research.","title":"Materials for a Course on Teaching Undergraduate Research","type":"resources"},{"content":" Summary This seminar series is designed to support success and professional development of students in science majors who are affiliated with the Iowa Sciences Academy.\nTaught During Spring 2025 as instructor of record ","date":"7 January 2025","externalUrl":null,"permalink":"/teaching/isa1040/","section":"Instruction","summary":"This seminar series is designed to support success and professional development of students in science majors who are affiliated with the Iowa Sciences Academy.","title":"ISA:1040 Entering Research","type":"teaching"},{"content":" Summary Scientific teaching principles (e.g., backwards design, active learning, formative assessment); students develop a teaching unit based on some aspect of their research and teach it to the class in preparation for future interviews where the ability to explain the background and significance for their research is a highly valued skill.\nTaught from Course Plan\nTaught During Fall 2024 as instructor of record ","date":"7 January 2025","externalUrl":null,"permalink":"/teaching/isa4040/","section":"Instruction","summary":"Scientific teaching principles (e.g., backwards design, active learning, formative assessment)","title":"ISA:4040 Teaching Your Undergraduate Research","type":"teaching"},{"content":" Summary Structure for development, planning, and implementation of a culminating project for the Iowa Sciences Academy; students dedicate three to five hours per week to the project and are encouraged to connect their projects to community issues or problem; integration of external learning experiences and activities including interviews, scientific observations, or internships\nTaught from Course Plan\nTaught During Spring 2025 as instructor of record ","date":"7 January 2025","externalUrl":null,"permalink":"/teaching/isa4041/","section":"Instruction","summary":"Structure for development, planning, and implementation of a culminating project for the Iowa Sciences Academy","title":"ISA:4041 Senior Capstone Project","type":"teaching"},{"content":" General Some Training Mature ","date":"13 March 2024","externalUrl":null,"permalink":"/what_is/knot/","section":"What is?","summary":"","title":"A Knot","type":"what_is"},{"content":" General Some Training Mature ","date":"13 March 2024","externalUrl":null,"permalink":"/what_is/tangle/","section":"What is?","summary":"","title":"A Tangle","type":"what_is"},{"content":" General Some Training Mature ","date":"13 March 2024","externalUrl":null,"permalink":"/what_is/tangle_tabulation/","section":"What is?","summary":"","title":"Tangle Tabulation","type":"what_is"},{"content":" Summary Topics in mathematical biology; canonical mathematical modeling and analysis of problems in the biological sciences; second of a two-semester sequence.\nCourse outline\nTaught During Spring 2024 as graduate assistant ","date":"7 January 2024","externalUrl":null,"permalink":"/teaching/math5760/","section":"Instruction","summary":"Topics in mathematical biology; canonical mathematical modeling and analysis of problems in the biological sciences; second of a two-semester sequence.","title":"MATH:5760 Mathematical Biology II (Module 6: Machine Learning)","type":"teaching"},{"content":" $\\newcommand{\\N}{\\mathbb{N}} \\newcommand{\\Z}{\\mathbb{Z}} \\newcommand{\\Q}{\\mathbb{Q}} \\newcommand{\\R}{\\mathbb{R}} \\newcommand{\\LP}{\\left(} \\newcommand{\\RP}{\\right)} \\newcommand{\\LS}{\\left\\lbrace} \\newcommand{\\RS}{\\right\\rbrace} \\newcommand{\\LA}{\\left\\langle} \\newcommand{\\RA}{\\right\\rangle} \\newcommand{\\LB}{\\left[} \\newcommand{\\RB}{\\right]} \\newcommand{\\MM}{\\ \\middle|\\ } \\newcommand{\\exp}{\\text{exp}} \\newcommand{\\abs}[1]{\\left\\vert#1\\right\\vert} \\newcommand{\\msr}[1]{m\\left(#1\\right)} \\newcommand{\\inv}[1]{#1^{-1}} \\newcommand{\\bkt}[1]{\\LA \\img{#1}\\RA} \\require{color}$ Rational Tangles A rational tangle is given by alternating NE,SE and SE,SW twisting of the $0$ tangle [1]. Discussion of canonicality of this construction of twist vector can be found in [2]. A twist vector encodes these alternating twists as a list of integers. This induces a unique map from the rational tangles onto the rational numbers [1][2][3]. We accomplish this by interpreting a twist vector as a sequence for a continued fraction as: $$\\LB a\\ b\\ c\\RB=c+\\frac{1}{b+\\frac{1}{a}}$$\nInstructions Twist vectors here are space separated lists of integers.\nA rational number here is \u0026ldquo;/\u0026rdquo; separating two integers.\nTwist Vector: Rational Number: J. Conway,\u0026#32;An enumeration of knots and links, and some of their algebraic properties,\u0026#32;in Computational Problems in Abstract Algebra,\u0026#32;Elsevier, 1970, pp. 329–358.doi:10.1016/B978-0-08-012975-4.50034-5J. Goldman and\u0026#32;L. Kauffman,\u0026#32;Rational Tangles,\u0026#32;Advances in Applied Mathematics,\u0026#32;vol. 18,\u0026#32;no. 3,\u0026#32;pp. 300–332,\u0026#32;1997.\u0026#32;doi:10.1006/aama.1996.0511L. Kauffman and\u0026#32;S. Lambropoulou,\u0026#32;On the Classification of Rational Knots,\u0026#32;arXiv: Geometric Topology,\u0026#32;2002.\u0026#32;doi:10.48550/ARXIV.MATH/0212011 ","date":"15 July 2023","externalUrl":null,"permalink":"/resources/tools/cont_frac_convert/","section":"Resources","summary":"A tool that computes continued fractions from twist vectors.","title":"Continued Fraction and Twist Vector Converter","type":"resources"},{"content":" PC Typesetting Core tool is vscode here are my settings\nplugins LaTeX Utilities LaTeX Workshop This is the main plugin: Here\u0026rsquo;s install guide Spell Checker gitlens free for students Todo Tree Bookmarks macros Settings \u0026#34;macros\u0026#34;: { \u0026#34;mathField\u0026#34;: [ { \u0026#34;command\u0026#34;: \u0026#34;type\u0026#34;, \u0026#34;args\u0026#34;: { \u0026#34;text\u0026#34;: \u0026#34;\\\\(\\\\)\u0026#34; } }, \u0026#34;cursorLeft\u0026#34;, \u0026#34;cursorLeft\u0026#34; ], \u0026#34;mathcField\u0026#34;: [ { \u0026#34;command\u0026#34;: \u0026#34;type\u0026#34;, \u0026#34;args\u0026#34;: { \u0026#34;text\u0026#34;: \u0026#34;\\\\[\\\\]\u0026#34; } }, \u0026#34;cursorLeft\u0026#34;, \u0026#34;cursorLeft\u0026#34; ], \u0026#34;mathField_jy\u0026#34;: [ { \u0026#34;command\u0026#34;: \u0026#34;type\u0026#34;, \u0026#34;args\u0026#34;: { \u0026#34;text\u0026#34;: \u0026#34;$$\u0026#34; } }, \u0026#34;cursorLeft\u0026#34; ], \u0026#34;mathcField_jy\u0026#34;: [ { \u0026#34;command\u0026#34;: \u0026#34;type\u0026#34;, \u0026#34;args\u0026#34;: { \u0026#34;text\u0026#34;: \u0026#34;$$$$\u0026#34; } }, \u0026#34;cursorLeft\u0026#34;, \u0026#34;cursorLeft\u0026#34; ], \u0026#34;mathField_md\u0026#34;: [ { \u0026#34;command\u0026#34;: \u0026#34;type\u0026#34;, \u0026#34;args\u0026#34;: { \u0026#34;text\u0026#34;: \u0026#34;\\\\\\\\(\\\\\\\\)\u0026#34; } }, \u0026#34;cursorLeft\u0026#34;, \u0026#34;cursorLeft\u0026#34;, \u0026#34;cursorLeft\u0026#34; ], \u0026#34;mathcField_md\u0026#34;: [ { \u0026#34;command\u0026#34;: \u0026#34;type\u0026#34;, \u0026#34;args\u0026#34;: { \u0026#34;text\u0026#34;: \u0026#34;\\\\\\\\[\\\\\\\\]\u0026#34; 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], \u0026#34;env\u0026#34;: {} } ], (Optional) Autobuild on push to trunk github action name: publish pdf on: push: branches: - main # Set a branch to deploy jobs: deploy: runs-on: ubuntu-latest steps: - name: Set up Git repository uses: actions/checkout@v2 - name: Compile LaTeX document uses: xu-cheng/latex-action@v2 with: latexmk_use_lualatex: true root_file: \u0026#34;*.tex\u0026#34; glob_root_file: true - name: Bump version and push tag id: tag_version uses: mathieudutour/github-tag-action@v6.0 with: github_token: ${{ secrets.GITHUB_TOKEN }} - name: Create a GitHub release uses: ncipollo/release-action@v1 with: tag: ${{ steps.tag_version.outputs.new_tag }} name: Release ${{ steps.tag_version.outputs.new_tag }} body: ${{ steps.tag_version.outputs.changelog }} - name: Release uses: softprops/action-gh-release@v1 with: tag_name: ${{ steps.tag_version.outputs.new_tag }} files: | *.pdf *.tex Development Package manager Using a package manager makes life a little easier here\u0026rsquo;s what I use on windows Choco.\nVersion Control choco install git.install -y For personal and academic work I use git with github for a production environment. I also maintain a local gogs server for local mirrors.\nI interact with local repositories with the vscode vs plugin. Rarely I use the comand line if vscode can\u0026rsquo;t do what I need. I\u0026rsquo;ve found gitkraken is a very accessable tool for new git users.\nFor professional work I use SVN.\nC/C++ Code editor I use vscode for editing code.\nDebugger I use visual studio 2022 for debugging.\nCompiler and build choco install ninja -y choco install cmake -y choco install llvm -y Set the following:\nCC=\u0026#34;C:\\Program Files\\LLVM\\bin\\clang.exe\u0026#34; CXX=\u0026#34;C:\\Program Files\\LLVM\\bin\\clang++.exe\u0026#34; Documentation engine/s Doxygen is basically the only option but I present the results with sphinx and breathe info here\nchoco install doxygen.install -y choco install sphinx -y pip install sphinx_rtd_theme pip install breathe Testing I use pytest tests run by cmake as ctest.\npip install pytest Python I use the latest lts python version. I use vscode as an editor and debugger. I use black for code formatting and flake8 for style checking.\nchoco install python -y pip install flake8 flake8-black scipy numpy pytest pytest-flake8 pytest-html pytest-metadata pytest-pylint pytest-reportlog flake8-docstrings flake8-html Academic Stuff Citations I use zotero for citation and pdf management between my ipad and windows pc.\nNotes I take notes with obsidian.\nPresentations Reveal.js Site\nDecktape Github\ndecktape -s 1920x1080 \u0026lt;url for presentation\u0026gt; presentation.pdf Ipad Zotero Site\nObsidian Site\nGoodnotes Site\nVectornator Site\nConcepts Site\nAnki Site\nQuick Install List vscode Choco python zotero visual studio 2022 gitkraken obsidian choco install ninja -y choco install cmake -y choco install llvm -y choco install doxygen.install -y choco install sphinx -y pip install sphinx_rtd_theme pip install breathe pip install pytest pip install flake8 flake8-black scipy numpy pytest pytest-flake8 pytest-html pytest-metadata pytest-pylint pytest-reportlog flake8-docstrings flake8-html For Tanglenomicon We\u0026rsquo;re still in a design phase here\u0026rsquo;s a list of DB options we\u0026rsquo;re considering:\nmongodb Apache cassandra choco install mongodb ","date":"20 June 2023","externalUrl":null,"permalink":"/resources/general/toolchain/","section":"Resources","summary":"","title":"Tool Chain","type":"resources"},{"content":" $\\newcommand{\\N}{\\mathbb{N}} \\newcommand{\\Z}{\\mathbb{Z}} \\newcommand{\\Q}{\\mathbb{Q}} \\newcommand{\\R}{\\mathbb{R}} \\newcommand{\\LP}{\\left(} \\newcommand{\\RP}{\\right)} \\newcommand{\\LS}{\\left\\lbrace} \\newcommand{\\RS}{\\right\\rbrace} \\newcommand{\\LA}{\\left\\langle} \\newcommand{\\RA}{\\right\\rangle} \\newcommand{\\LB}{\\left[} \\newcommand{\\RB}{\\right]} \\newcommand{\\MM}{\\ \\middle|\\ } \\newcommand{\\exp}{\\text{exp}} \\newcommand{\\abs}[1]{\\left\\vert#1\\right\\vert} \\newcommand{\\msr}[1]{m\\left(#1\\right)} \\newcommand{\\inv}[1]{#1^{-1}} \\newcommand{\\bkt}[1]{\\LA \\img{#1}\\RA} \\require{color}$ Instructions Create a $n\\times n$ table by setting the size and pressing the button. Select the colors for your knot. Select the scale, size of the strands, for your knot. Select the eccentricity, how much the control points pull, for your knot. Fill in the table with the grid diagram. For info on using grid diagrams to define Legendrian knots see 10.48550/arXiv.1903.12256 or 10.2140/agt.2010.10.293. Note there is no error handling so if something goes wrong refresh the page and try again Click generate to get the image. Click \u0026ldquo;Download\u0026rdquo; to download a copy of the image. Size : Create table Foreground color:\nBackground color:\nEccentricity :\nScale :\nConvert table Download Sample images x o x o x o x o o x x o x o x o x o x o x x o x o x o o x x o o x x o x o o x x o o x x o o x ","date":"30 May 2023","externalUrl":null,"permalink":"/resources/tools/grid_to_front_projection/","section":"Resources","summary":"","title":"Grid Diagram to Front Projection","type":"resources"},{"content":" Summary Algebraic techniques, equations and inequalities, functions and graphs, exponential and logarithmic functions, systems of equations and inequalities.\nTaught from ALEKS Taught During Fall 2022 as instructor of record Fall 2023 as instructor of record ","date":"7 January 2023","externalUrl":null,"permalink":"/teaching/math1005/","section":"Instruction","summary":"College Algebra","title":"MATH:1005 College Algebra","type":"teaching"},{"content":" Summary Algebraic techniques and modeling; quantitative methods for treating problems that arise in management and economic sciences; topics include algebra techniques, functions and functional models, exponential and logarithmic functions and models, and a thorough introduction to differential calculus; examples and applications from management, economic sciences, and related areas; for students planning to major in business.\nTaught from Lial, Hungerford, Holcomb, \u0026amp; Mullins: Mathematics with Applications in the Management, Natural, and Social Sciences, 12th Edition. Taught During Fall 2021 as graduate assistant Spring 2022 as graduate assistant Spring 2024 as graduate assistant ","date":"7 January 2023","externalUrl":null,"permalink":"/teaching/math1350/","section":"Instruction","summary":"Mathematics and Reasoning for business students","title":"MATH:1350 Quantitative Reasoning for Business","type":"teaching"},{"content":"A prepsheet/card deck for preparing for my analysis qual\nhttps://github.com/Joecstarr/Analysis_prepsheet\nTeX Generates to a pdf targeting an ipad. Pdf has one page of definition name followed by a page of the actual definition content.\nAnki anki.py is a python script, it takes in a .tex file (drag and drop). It then parses out the review content and creates an anki deck. Import the deck into anki desktop to review with ankiweb.\n","date":"7 January 2020","externalUrl":null,"permalink":"/resources/general/analysis_prepsheet/","section":"Resources","summary":"","title":"Analysis Prepsheet","type":"resources"},{"content":"Rubber duck debugging is a playful yet effective method used by programmers to troubleshoot code. The concept involves explaining your code, line by line, to an inanimate object—often a rubber duck. By verbalizing the problem, you can uncover errors or gain new insights that might not be apparent when simply reading the code. This technique leverages the cognitive benefits of teaching and articulation, helping to clarify your thoughts and identify issues. It\u0026rsquo;s a testament to the power of simplicity in problem-solving, proving that sometimes, the best solutions come from the most unexpected sources. So, next time you\u0026rsquo;re stuck on a tricky bug, grab a rubber duck and start talking!\nThis is my collection of rubber ducks, currently standing at ducks.\nPhoto credit Valeria Reyna.\nDuck of the day! Ducks on the pond! ","date":"7 January 2020","externalUrl":null,"permalink":"/misc/ducks/","section":"","summary":"","title":"Rubber Ducks","type":"misc"},{"content":" $\\newcommand{\\N}{\\mathbb{N}} \\newcommand{\\Z}{\\mathbb{Z}} \\newcommand{\\Q}{\\mathbb{Q}} \\newcommand{\\R}{\\mathbb{R}} \\newcommand{\\LP}{\\left(} \\newcommand{\\RP}{\\right)} \\newcommand{\\LS}{\\left\\lbrace} \\newcommand{\\RS}{\\right\\rbrace} \\newcommand{\\LA}{\\left\\langle} \\newcommand{\\RA}{\\right\\rangle} \\newcommand{\\LB}{\\left[} \\newcommand{\\RB}{\\right]} \\newcommand{\\MM}{\\ \\middle|\\ } \\newcommand{\\exp}{\\text{exp}} \\newcommand{\\abs}[1]{\\left\\vert#1\\right\\vert} \\newcommand{\\msr}[1]{m\\left(#1\\right)} \\newcommand{\\inv}[1]{#1^{-1}} \\newcommand{\\bkt}[1]{\\LA \\img{#1}\\RA} \\require{color}$ Defense Slides (8/18/25) The Tanglenomicon: Tabulation of Arborescent Tangles Joseph Starr Mathematics Department at The University of Iowa Partially supported by DMS-2038103 \"The Tanglenomicon\" name due to Dr. Nicholas Connolly Knots $\\quad$ $\\quad$ $\\quad$ https://www.knotplot.com/\nThe natural question How many knots? Knot Tables Lord Kelvin\u0026rsquo;s vortex theory of the atom Atoms are knotted vortices in the æther. By Hand 1860s Tait computes knots up to 7 crossings 15 knots 1870s Tait, Kirkman, and Little compute knots up to 10 crossings Takes about 25 years 250 knots (+1 repeat in the Perko pair) 1960s Conway computes knots up to 11 crossings \u0026ldquo;A few hours\u0026rdquo; 1980s Caudron verifies knots up to 11 crossings Finding 4 omissions of Conway By Computer 1980s Dowker and Thistlethwaite compute up to 13 crossings First using a computer 12,966 knots 1990s Hoste, Thistlethwaite, and Weeks compute up to 16 crossings Computer runtime on the order of weeks 1,701,936 knots 2020s Burton computes up to 19 crossings Computer runtime on the order of months 350 million knots Conway How did Conway compute 25 years of work in \"a few hours\"? Tangles \u0026ldquo;We define a tangle as a portion of a knot diagram from which there emerge just 4 arcs pointing in the compass directions NW, NE, SW, SE.\u0026rdquo; - Conway, J.H.\nConway, J.H. \u0026ldquo;An Enumeration of Knots and Links, and Some of Their Algebraic Properties.\u0026rdquo; In Computational Problems in Abstract Algebra, 329-58. Elsevier, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5\n$\\quad$ $\\quad$ NWNESWSE $\\quad$ $\\quad$ $\\quad$ Basic Operations Operation $+$ $+$ $=$ $=$ $=$ $2$ Operation $\\vee$ $\\vee$ $=$ $=$ $=$ $\\frac{1}{2}$ Algebraic Tangles All possible tangles made from $+$ and $\\vee$ on basic tangles.\nAlgebraic A tangle built from $\\vee$ and $+$ on some basic tangles. $$\\color{var(--r-foreground)}\\LP\\color{var(--r-Purple)} \\LP\\LP3\\vee\\\\\\frac{1}{2}\\RP+3\\RP+\\LP\\LP3\\vee\\frac{1}{2}\\RP+3\\RP\\color{var(--r-foreground)}\\RP \\vee\\color{var(--r-foreground)}\\LP\\color{var(--r-Purple)} \\LP\\LP3\\vee\\frac{1}{2}\\RP+3\\RP+\\LP\\LP3\\vee\\frac{1}{2}\\RP+3\\RP\\color{var(--r-foreground)}\\RP $$ $$\\vee \\color{var(--r-Purple)}++\\vee3\\frac{1}{2}3+\\vee3\\frac{1}{2}3 ++\\vee3\\frac{1}{2}3+\\vee3\\frac{1}{2}3 $$ Conway, J.H. \u0026ldquo;An Enumeration of Knots and Links, and Some of Their Algebraic Properties.\u0026rdquo; In Computational Problems in Abstract Algebra, 329-58. Elsevier, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5\nArborescent Tangles Arborescent knots (and tangles) are constructed by taking a collection of twisted bands described by a weighted tree and connecting them with non-cyclic successive plumbing.\nAlgebraic and arborescent constructions describe the same class of objects.\nF. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nx Y x Y \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e $\\ $ We can see here the correspondence between algebraic and arborescent constructions.\nHow to encode? (Bonahon and Siebenmann) Relationship between a band and the other bands plumbed to it (children). Location of twists relative to the children. Tangle boundary. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\n1. Relationship between a band and the other bands attached to it (children) Acyclic connections between items. Relative positions of connections. Acyclic connections between items? Solution: A tree (in the graph sense) \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e Relative positions of connections? Solution: Rooted plane tree Definition A rooted plane tree is an abstract tree imbued with a strict total order on the vertices (indexed by the non-negative integers). We call the least vertex the root of the tree. Convention: We will select the total order given by a depth first in order traversal of the tree. Cyclic Order of a Vertex C0 C2 C1 $\\quad$ C2 C0 C1 \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e How to encode? (Bonahon and Siebenmann) ✓ Relationship between a band and the other bands attached to it (children). Location of twists relative to the children. Tangle boundary. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\n\u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e -2 W0 W2 W1 C0 C2 C1 3 2 -3 0 4 3 $\\quad$ How to encode? (Bonahon and Siebenmann) ✓ Relationship between a band and the other bands attached to it (children). ✓ Location of twists relative to the children. Tangle boundary. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nLocal view of a vertex Bonds \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e W0 W2 W1 Free Bonds $\\ $ \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e 3 2 -3 0 4 {ι,ξ,ς,η} Rotations Elements of Klein four-group ($V_4$) $\\ $ Y How to encode? (Bonahon and Siebenmann) ✓ Relationship between a band and the other bands attached to it (children). ✓ Location of twists relative to the children. ✓ Tangle boundary. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nWeighted planar tangle tree \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e 3 2 -3 0 4 {ι,ξ,ς,η} Anatomy of a Weighted planar tangle tree Rings 2 1 2 -2 -1 -2 $\\ $ F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\n3 2 -3 4 3 Definition A vertex with valence $\\geq 3$ is called an Essential Vertex. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\n3 2 -3 4 3 Definition The subtrees remaining after excising all essential vertices and their bonds (half edges) are called the Sticks of a tree. 3 -3 4 3 F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\n\u0026#953; 3 3 -2 3 -2 Moves on a Weighted Planar Tangle Tree Definition: Part 1 The $F_3^\\prime$ move on a weighted arborescent tree moves a weight $w$ as in the image below and, if $w$ is odd, reverse the cyclic order of weights and bonds at all vertices of the purple subtree lying at odd distance (count of edges between two vertices) from the vertex shown. w w w F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nThe flype \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u0026ldquo;if $w$ is odd, reverse the cyclic order of weights and bonds at all vertices of the purple subtree lying at odd distance (count of edges between two vertices) from the vertex shown.\u0026rdquo;\n\u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e 1 1 \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e 2 2 1 1 Definition: Part 2 Also, when $w$ is odd, apply $\\xi$ ( $X$-axis rotation) to all free bonds in the purple subtree that are attached to a vertex at even distance from the vertex shown, and $\\eta$ ($Y$-axis rotation) to those at odd distance. The rotations are relative to the local orientations of the plumbing squares on the bands corresponding to vertices at even/odd distances from the vertex with weight $w$. w w w F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\n\u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e Definition The $F_2$ move on a weighted arborescent tangle tree reverses the cyclic order of bonds and weights at one vertex on the tree and at every vertex at even distance from it; also apply $\\eta$ ($Y$-axis rotation) to every free bond of a vertex at even (or zero) distance, and apply $\\xi$ ($X$-axis rotation) to every free bond at odd distance. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nDefinition The $F_1$ move on a weighted arborescent tangle tree reverses the cyclic order of bonds and weights at every vertex of the graph and applies $\\zeta$ ($Z$-axis rotation) to every free bond. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nDefinition The $R^-$ replaces the left image below with the right, leaving the rest of the tree unchanged. -2 -2 -1 -2 -2 -1 F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nDefinition The $R^+$ replaces the left image below with the right, leaving the rest of the tree unchanged. 2 2 1 2 2 1 F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\n\u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e Canonical Trees Canonical Trees For our purposes, a weighted planar tree $\\Gamma$ is called a canonical weighted planar tangle tree (CWPTT) if it has a single free bond with a label from $V_4$ and satisfies the following conditions.\nF. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nWeight Condition At each vertex of $\\Gamma$, at most one weight is non-zero.\nF. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nStick Conditions On any stick the weights of the vertices are non-zero except for end vertices that have a bond free in $\\Gamma$ and for the case $\\Gamma$ is: \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u0026lt;sodipodi:namedview id=\u0026ldquo;namedview3\u0026rdquo; pagecolor=\u0026quot;#ffffff\u0026quot; bordercolor=\u0026quot;#000000\u0026quot; borderopacity=\u0026ldquo;0.25\u0026rdquo; inkscape:showpageshadow=\u0026ldquo;2\u0026rdquo; inkscape:pageopacity=\u0026ldquo;0.0\u0026rdquo; inkscape:pagecheckerboard=\u0026ldquo;0\u0026rdquo; inkscape:deskcolor=\u0026quot;#d1d1d1\u0026quot; inkscape:zoom=\u0026ldquo;1.4972695\u0026rdquo; inkscape:cx=\u0026ldquo;202.36838\u0026rdquo; inkscape:cy=\u0026ldquo;231.42126\u0026rdquo; inkscape:window-width=\u0026ldquo;2234\u0026rdquo; inkscape:window-height=\u0026ldquo;1418\u0026rdquo; inkscape:window-x=\u0026ldquo;0\u0026rdquo; inkscape:window-y=\u0026ldquo;0\u0026rdquo; inkscape:window-maximized=\u0026ldquo;1\u0026rdquo; inkscape:current-layer=\u0026ldquo;svg3\u0026rdquo; /\u0026gt; or \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u0026lt;sodipodi:namedview id=\u0026ldquo;namedview5\u0026rdquo; pagecolor=\u0026quot;#ffffff\u0026quot; bordercolor=\u0026quot;#000000\u0026quot; borderopacity=\u0026ldquo;0.25\u0026rdquo; inkscape:showpageshadow=\u0026ldquo;2\u0026rdquo; inkscape:pageopacity=\u0026ldquo;0.0\u0026rdquo; inkscape:pagecheckerboard=\u0026ldquo;0\u0026rdquo; inkscape:deskcolor=\u0026quot;#d1d1d1\u0026quot; inkscape:zoom=\u0026ldquo;1.7833534\u0026rdquo; inkscape:cx=\u0026ldquo;98.410109\u0026rdquo; inkscape:cy=\u0026ldquo;350.18298\u0026rdquo; inkscape:window-width=\u0026ldquo;2234\u0026rdquo; inkscape:window-height=\u0026ldquo;1418\u0026rdquo; inkscape:window-x=\u0026ldquo;0\u0026rdquo; inkscape:window-y=\u0026ldquo;0\u0026rdquo; inkscape:window-maximized=\u0026ldquo;1\u0026rdquo; inkscape:current-layer=\u0026ldquo;svg5\u0026rdquo; /\u0026gt; The non-zero weights along any stick are of alternating sign. No end vertex of a stick has weight $\\pm 1$ unless it has a bond free in $\\Gamma$. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nOne of Positivity Condition $\\LP+\\RP$-CWPTT Except for those with a free bond, there are no sticks in $\\Gamma$ of the forms \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u0026lt;sodipodi:namedview id=\u0026ldquo;namedview4\u0026rdquo; pagecolor=\u0026quot;#ffffff\u0026quot; bordercolor=\u0026quot;#000000\u0026quot; borderopacity=\u0026ldquo;0.25\u0026rdquo; inkscape:showpageshadow=\u0026ldquo;2\u0026rdquo; inkscape:pageopacity=\u0026ldquo;0.0\u0026rdquo; inkscape:pagecheckerboard=\u0026ldquo;0\u0026rdquo; inkscape:deskcolor=\u0026quot;#d1d1d1\u0026quot; inkscape:zoom=\u0026ldquo;1.941329\u0026rdquo; inkscape:cx=\u0026ldquo;67.479548\u0026rdquo; inkscape:cy=\u0026ldquo;149.12465\u0026rdquo; inkscape:window-width=\u0026ldquo;2254\u0026rdquo; inkscape:window-height=\u0026ldquo;1502\u0026rdquo; inkscape:window-x=\u0026ldquo;0\u0026rdquo; inkscape:window-y=\u0026ldquo;0\u0026rdquo; inkscape:window-maximized=\u0026ldquo;1\u0026rdquo; inkscape:current-layer=\u0026ldquo;svg4\u0026rdquo; /\u0026gt; or \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u0026lt;sodipodi:namedview id=\u0026ldquo;namedview5\u0026rdquo; pagecolor=\u0026quot;#ffffff\u0026quot; bordercolor=\u0026quot;#000000\u0026quot; borderopacity=\u0026ldquo;0.25\u0026rdquo; inkscape:showpageshadow=\u0026ldquo;2\u0026rdquo; inkscape:pageopacity=\u0026ldquo;0.0\u0026rdquo; inkscape:pagecheckerboard=\u0026ldquo;0\u0026rdquo; inkscape:deskcolor=\u0026quot;#d1d1d1\u0026quot; inkscape:zoom=\u0026ldquo;1.7818863\u0026rdquo; inkscape:cx=\u0026ldquo;107.75098\u0026rdquo; inkscape:cy=\u0026ldquo;480.3898\u0026rdquo; inkscape:window-width=\u0026ldquo;2254\u0026rdquo; inkscape:window-height=\u0026ldquo;1502\u0026rdquo; inkscape:window-x=\u0026ldquo;0\u0026rdquo; inkscape:window-y=\u0026ldquo;0\u0026rdquo; inkscape:window-maximized=\u0026ldquo;1\u0026rdquo; inkscape:current-layer=\u0026ldquo;svg5\u0026rdquo; /\u0026gt; or Negativity Condition $\\LP-\\RP$-CWPTT Except for those with a free bond, there are no sticks in $\\Gamma$ of the forms \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u0026lt;sodipodi:namedview id=\u0026ldquo;namedview4\u0026rdquo; pagecolor=\u0026quot;#ffffff\u0026quot; bordercolor=\u0026quot;#000000\u0026quot; borderopacity=\u0026ldquo;0.25\u0026rdquo; inkscape:showpageshadow=\u0026ldquo;2\u0026rdquo; inkscape:pageopacity=\u0026ldquo;0.0\u0026rdquo; inkscape:pagecheckerboard=\u0026ldquo;0\u0026rdquo; inkscape:deskcolor=\u0026quot;#d1d1d1\u0026quot; inkscape:zoom=\u0026ldquo;4.4031681\u0026rdquo; inkscape:cx=\u0026ldquo;148.98364\u0026rdquo; inkscape:cy=\u0026ldquo;241.07642\u0026rdquo; inkscape:window-width=\u0026ldquo;2254\u0026rdquo; inkscape:window-height=\u0026ldquo;1502\u0026rdquo; inkscape:window-x=\u0026ldquo;0\u0026rdquo; inkscape:window-y=\u0026ldquo;0\u0026rdquo; inkscape:window-maximized=\u0026ldquo;1\u0026rdquo; inkscape:current-layer=\u0026ldquo;svg4\u0026rdquo; /\u0026gt; or \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u0026lt;sodipodi:namedview id=\u0026ldquo;namedview5\u0026rdquo; pagecolor=\u0026quot;#ffffff\u0026quot; bordercolor=\u0026quot;#000000\u0026quot; borderopacity=\u0026ldquo;0.25\u0026rdquo; inkscape:showpageshadow=\u0026ldquo;2\u0026rdquo; inkscape:pageopacity=\u0026ldquo;0.0\u0026rdquo; inkscape:pagecheckerboard=\u0026ldquo;0\u0026rdquo; inkscape:deskcolor=\u0026quot;#d1d1d1\u0026quot; inkscape:zoom=\u0026ldquo;0.94705423\u0026rdquo; inkscape:cx=\u0026ldquo;122.48507\u0026rdquo; inkscape:cy=\u0026ldquo;63.882297\u0026rdquo; inkscape:window-width=\u0026ldquo;2254\u0026rdquo; inkscape:window-height=\u0026ldquo;1502\u0026rdquo; inkscape:window-x=\u0026ldquo;0\u0026rdquo; inkscape:window-y=\u0026ldquo;0\u0026rdquo; inkscape:window-maximized=\u0026ldquo;1\u0026rdquo; inkscape:current-layer=\u0026ldquo;svg5\u0026rdquo; /\u0026gt; F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nTheorem: Bonahon and Seibenmann There exists an effective algorithm which, for any weighted planar tree $\\Gamma$ with free bonds labeled by elements of $V_4$, alters $\\Gamma$ by a sequence of moves of the calculus of arborescent tangles to produce a collection of positively (or negatively) canonical weighted planar trees. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nTheorem: Bonahon and Seibenmann Consider two positive (or negative) CWPTT $\\Gamma$ and $\\Gamma^\\prime$, with free bonds labeled by elements of $V_4$. Plumbing, according to $\\Gamma$ and $\\Gamma^\\prime$ gives isomorphic arborescent tangles if and only if $\\Gamma$ and $\\Gamma^\\prime$ can be deduced from each other by a sequence of moves ($F_1$), ($F_2$), ($F_3^\\prime$), and the modified ring moves $\\LP\\pm R\\RP$. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nDefinition For weighted planar tangle tree $\\Gamma$, with weights $\\LS w_i\\RS$. We call $$\\text{TCN}=\\sum |w_i|$$ the Tree Crossing Number (TCN). CWPTT are Not Minimal 2 -2 -2 2 $$\\to$$ 2 2 2 2 2 A Canonical Vertex [S] A vertex $v_i$ of a weighted planar tangle tree $\\Gamma$ with a single free bond labeled from $V_4$ is said to be $\\LP+\\RP$-canonical if $v_i$ has at most one non-zero weight $w_i$ and $i$ is:\nZero ($v_0$ the root) with no further conditions. Non-zero with the following conditions satisfied: I. If the valence of $v_i$ is $1$ then all of:\n(Stick Condition) $w_i\\neq 0$ unless $i=1$ and $w_0=0$ (the weight of the root) $w_i\\neq \\pm 1$ If the valence of $v_{i-1}$ (the parent) is $2$ then $\\text{sign}\\LP w_i\\RP\\neq\\text{sign}\\LP w_{i-1}\\RP$ unless $i=1$ and $w_0=0$ (Positivity Condition) If the valence of $v_{i-1}$ (the parent) is greater than $2$ then $w_i\\neq -2$ Stick Condition 1. On any stick the weights of the vertices are non-zero except for end vertices that have a bond free in $\\Gamma$ and for the case $\\Gamma$ is: \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e $$\\text{or}$$ \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e 2. The non-zero weights along any stick are of alternating sign. 3. No end vertex of a stick has weight $\\pm 1$ unless it has a bond free in $\\Gamma$. I. If the valence of $v_i$ is $1$ then all of:\n(Stick Condition) $w_i\\neq 0$ unless $i=1$ and $w_0=0$ (the weight of the root) $w_i\\neq \\pm 1$ If the valence of $v_{i-1}$ (the parent) is $2$ then $\\text{sign}\\LP w_i\\RP\\neq\\text{sign}\\LP w_{i-1}\\RP$ unless $i=1$ and $w_0=0$ Positivity Condition: Except for those with a free bond, there are no sticks in $\\Gamma$ of the forms \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e $$\\text{or}$$ \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e I. If the valence of $v_i$ is $1$ then all of:\n(Positivity Condition) If the valence of $v_{i-1}$ (the parent) is greater than $2$ then $w_i\\neq -2$ II. If the valence of $v_i$ is $2$ then all of:\n(Stick Condition) $w_i\\neq 0$ If valence of the parent or valence of the child is greater than $2$ (is essential) then $w_i\\neq \\pm1$ $\\ $ If valence of the parent is $2$ then $\\text{sign}\\LP w_i\\RP\\neq\\text{sign}\\LP w_{i-1}\\RP$ (the parent) If valence of the child is $2$ then $\\text{sign}\\LP w_i\\RP\\neq\\text{sign}\\LP w_{i+1}\\RP$ (the child) (Positivity Condition) If valence of the parent and valence of the child is greater than $2$ (both are essential) then $w_i\\neq -2$ Stick Condition 1. On any stick the weights of the vertices are non-zero except for end vertices that have a bond free in $\\Gamma$ and for the case $\\Gamma$ is: \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e $$\\text{or}$$ \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e 2. The non-zero weights along any stick are of alternating sign. 3. No end vertex of a stick has weight $\\pm 1$ unless it has a bond free in $\\Gamma$. II. If the valence of $v_i$ is $2$ then all of:\n(Stick Condition) $w_i\\neq 0$ If valence of the parent or valence of the child is greater than $2$ (is essential) then $w_i\\neq \\pm1$ $\\ $ If valence of the parent is $2$ then $\\text{sign}\\LP w_i\\RP\\neq\\text{sign}\\LP w_{i-1}\\RP$ (the parent) If valence of the child is $2$ then $\\text{sign}\\LP w_i\\RP\\neq\\text{sign}\\LP w_{i+1}\\RP$ (the child) Positivity Condition: Except for those with a free bond, there are no sticks in $\\Gamma$ of the forms \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e $$\\text{or}$$ \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e II. If the valence of $v_i$ is $2$ then all of:\n(Positivity Condition) If valence of the parent and valence of the child is greater than $2$ (both are essential) then $w_i\\neq -2$ Theorem [S] $\\Gamma$ is a $\\LP+\\RP$-CWPTT if and only if all the vertices of $\\Gamma$ are $\\LP+\\RP$-canonical. Tabulation of Arborescent Tangles What do we have? A combinatorial structure A combinatorial structure A way to distinguish two arborescent tangles What we don\u0026rsquo;t have? A unique representative A unique representative An efficient storage strategy A unique representative An efficient storage strategy An efficient generation strategy A unique representative Definition [S] A CWPTT is called right leaning if all weights are in the highest indexed region of each vertex. Additionally, any ring subtrees that are children of a vertex are the highest indexed children of that vertex. Note Not using \u0026ldquo;abbreviated trees\u0026rdquo; makes generation a bit less complex. W0 W2 W1 C0 C2 C1 Theorem [S] Every arborescent tangle has a right leaning CWPTT representative. Definition [S] A CWPTT is called an identity tree if its free bond is marked by $\\iota\\in V_4$. Theorem [S] Every arborescent tangle has an identity CWPTT representative. Definition [S] A CWPTT is called a right leaning identity tangle tree (RLITT) if it\u0026rsquo;s a right leaning and identity tree. Theorem [S] Every $\\LP+\\RP$-CWPTT has a unique right leaning identity representative. ✓ A unique representative An efficient storage strategy An efficient generation strategy An efficient storage strategy \u0026#953;((2[3][-3])(((([3][4])[8][2 1])-2)2)) \u0026#953; Cayley, A. (1857). ON THE THEORY OF THE ANALYTICAL FORMS CALLED TREES (The collected mathematical papers Vol. 3). Cambridge University Press. https://doi.org/10.1017/CBO9780511703690\n✓ A unique representative ✓ An efficient storage strategy An efficient generation strategy An efficient generation strategy Definition [S] For WPTT $\\Gamma_r$ and $\\Gamma_s$, define the grafting operation $$ \\begin{aligned} \\Gamma_r\\times\\Gamma_s\u0026amp;\\mapsto\\Gamma_r\\star_i\\Gamma_s \\end{aligned} $$ as follows. At the vertex $v_i$ of a $\\Gamma_r$, introduce a bond connecting to the free bond at the root of $\\Gamma_s$, reindexing as needed. We also require that $\\Gamma_s$ be grafted so that the rightmost weight of $v_i$ and any ring subtrees of $v_i$ remain to the right of the scion after grafting.\nWhen grafting at the root $v_0$ we omit the index label in the grafting operation, that is, $\\star_0$ is written simply as $\\star$. We call $\\Gamma_r$ the rootstock and $\\Gamma_s$ the scion.\n$$\\Gamma_r\\text{ and }\\color{#ffb86c}\\Gamma_s$$\n\u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e 2 1 2 $$\\Gamma_r\\star_2\\color{#ffb86c}\\Gamma_s$$\n\u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e 2 1 2 Theorem [S] Every $\\Gamma$ $\\LP+\\RP$-RLITT of TCN $n$ is one of two forms:\n$\\Gamma$ is a single vertex with weight $\\pm n$. $\\Gamma$ is the result of grafting at the root of some rootstock $\\Gamma_r$ and $\\LP+\\RP$-RLITT scion $\\Gamma_s$ where: In $\\Gamma_r$, $v_0$ is valence two, and $v_1$ is canonical except for violating the stick condition by $\\text{Sign}\\LP v_0\\RP=\\text{Sign}\\LP v_1\\RP$. Each vertex in $\\LS v_i\\RS_{i=2}^n$ of $\\Gamma_r$ is $\\LP +\\RP$-canonical. $\\Gamma_r$ is $\\LP+\\RP$-RLITT \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e -2 3 3 1 \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e -2 3 3 -1 Algorithm [S] Input\nA collection of of RLITT scions $T_s$ A collection of RLITT rootstocks $T_r$ Output\nA collection of weighted planar trees Routine\nfor each combination of $\\Gamma_r\\in T_r$ and $\\Gamma_s \\in T_s$ Compute $\\Gamma = \\Gamma_r\\star \\Gamma_s$ Report $\\Gamma$ Continue to the iteration of the loop if $v_0$ in $\\Gamma_r$ is valence other than two Set $v_0$ to $-v_0$ for $\\Gamma$ Report $\\Gamma$ Guarantee RLITT Weight Condition Identity Condition Right Leaning Condition Stick Condition Positivity/Negativity Condition Guarantee RLITT ✓ Weight Condition ✓ Identity Condition ✓ Right Leaning Condition Stick Condition Positivity/Negativity Condition Stick Condition Theorem [S] For $\\Gamma_r$ a $\\LP+\\RP-$RLITT and $\\Gamma_s$ a $\\LP+\\RP-$RLITT scion, the result of $\\Gamma=\\Gamma_r\\star\\Gamma_s$ is canonical, if all $v_i$ at distance $1$ or less from the root are canonical. Definition [S] A $\\LP+\\RP-$RLITT (respectively $\\LP-\\RP-$RLITT) $\\Gamma$ with root weight $w_0$ is called a good scion when either:\n$w_0\\neq0$ $w_0=0$ and the valence of $v_0$ is greater than $2$ (essential) Algorithm [S] Input\nA collection of RLITT good scions $T_s$ A collection of RLITT rootstocks $T_r$ Output\nA collection of weighted planar trees (still not guaranteed to be RLITT) Routine\nfor each combination of $\\Gamma_s\\in T_s$ and $\\Gamma_r \\in T_r$ Compute $\\Gamma = \\Gamma_r\\star \\Gamma_s$ for each vertex $v_i$ at distance 1 from the root of $\\Gamma$ Continue to the next iteration of the outer loop if $v_i$ fails to satisfy the stick condition Report $\\Gamma$ Continue to the iteration of the loop if $v_0$ in $\\Gamma_r$ is valence other than two Set $v_0$ to $-v_0$ for $\\Gamma$ Report $\\Gamma$ Guarantee RLITT ✓ Weight Condition ✓ Identity Condition ✓ Right Leaning Condition ✓ Stick Condition Positivity/Negativity Condition Positivity/Negativity Condition Theorem [S] For $\\Gamma_r$ a non-negative $\\LP+\\RP-$RLITT, and $\\Gamma_s$ a good non-positive $\\LP-\\RP-$RLITT scion, the result of $\\Gamma=\\Gamma_r\\star\\Gamma_s$ is non-canonical. Algorithm: Generate $\\LP\u0026#43;\\RP$-RLITT [S] Input\nA collection of $\\LP+\\RP$-RLITT good scions $T_s$ A collection of $\\LP+\\RP$-RLITT rootstocks $T_r$ Output\nA collection of weighted planar trees (still not guaranteed to be RLITT) Routine\nfor each combination of $\\Gamma_s\\in T_s$ and $\\Gamma_r \\in T_r$ Compute $\\Gamma = \\Gamma_r\\star \\Gamma_s$ for each vertex $v_i$ at distance 1 from the root of $\\Gamma$ Continue to the next iteration of the outer loop if $v_i$ fails to satisfy the stick condition Continue to the next iteration of the outer loop if $v_i$ fails to satisfy the positivity condition Report $\\Gamma$ Continue to the iteration of the loop if $v_0$ in $\\Gamma_r$ is valence other than two Set $v_0$ to $-v_0$ for $\\Gamma$ Report $\\Gamma$ Algorithm: Generate $\\LP-\\RP$-RLITT [S] Input\nA collection of $\\LP-\\RP$-RLITT good scions $T_s$ A collection of $\\LP-\\RP$-RLITT rootstocks $T_r$ Output\nA collection of weighted planar trees (still not guaranteed to be RLITT) Routine\nfor each combination of $\\Gamma_s\\in T_s$ and $\\Gamma_r \\in T_r$ Compute $\\Gamma = \\Gamma_r\\star \\Gamma_s$ for each vertex $v_i$ at distance 1 from the root of $\\Gamma$ Continue to the next iteration of the outer loop if $v_i$ fails to satisfy the stick condition Continue to the next iteration of the outer loop if $v_i$ fails to satisfy the positivity condition Report $\\Gamma$ Continue to the iteration of the loop if $v_0$ in $\\Gamma_r$ is valence other than two Set $v_0$ to $-v_0$ for $\\Gamma$ Report $\\Gamma$ Guarantee RLITT ✓ Weight Condition ✓ Identity Condition ✓ Right Leaning Condition ✓ Stick Condition ✓ Positivity/Negativity Condition Partitioning jobs $$\\begin{aligned} (0\u0026amp;,\\text{TCN})\\\\ ( 1\u0026amp;,\\text{TCN}-1)\\\\ \u0026amp;\\vdots\\\\ (\\text{TCN}-1\u0026amp;,1)\\\\ (\\text{TCN}\u0026amp;,0) \\end{aligned} $$\nAlgorithm [S] Input\nA target TCN $n$ Output\nA set $T$ of all RLITT up to TCN Routine\nSet $T$ to be the set $\\LS \\iota[0],\\ \\iota[0\\ 0],\\ \\iota[1],\\ \\iota[-1],\\ \\iota[2],\\ \\iota[-2],\\ \\cdots,\\ \\iota[n],\\ \\iota[-n],\\ \\RS$ for i from 2 to TCN for j from i-2 to 1 Set $T_{r+}$ to be the set of $(+)$-RLITT with TCN equal to $j$ Set $T_{r-}$ to be the set of $(-)$-RLITT with TCN equal to $j$ Set $T_{s+}$ to be the set of $(+)$-RLITT good scions with TCN equal to $n-j$ Set $T_{s-}$ to be the set of $(-)$-RLITT good scions with TCN equal to $n-j$ Execute \u0026ldquo;Generate $\\LP+\\RP$-RLITT\u0026rdquo; input $T_{r+}$ and $T_{s+}$ Execute \u0026ldquo;Generate $\\LP-\\RP$-RLITT\u0026rdquo; input $T_{r-}$ and $T_{s-}$ Set $T_{r+}$ to be the set of $(+)$-RLITT with TCN equal to $0$ Set $T_{r-}$ to be the set of $(-)$-RLITT with TCN equal to $0$ Set $T_{s+}$ to be the set of $(+)$-RLITT good scions with TCN equal to $i$ Set $T_{s-}$ to be the set of $(-)$-RLITT good scions with TCN equal to $i$ Execute \u0026ldquo;Generate $\\LP+\\RP$-RLITT\u0026rdquo; input $T_{r+}$ and $T_{s+}$ Execute \u0026ldquo;Generate $\\LP-\\RP$-RLITT\u0026rdquo; input $T_{r-}$ and $T_{s-}$ Add the results to $T$ Technologies ThrowTheSwitch/Unity Simple Unit Testing for C C 3.3k 935 Future work Minimalization of Arborescent representatives 2 -2 -2 2 $$\\leftarrow$$ 2 2 2 2 2 Minimalization general representatives Tabulation of Polygonal tangles Algebraic $+$, $\\vee$, and plumbing only form bigons between basic tangles in the \u0026ldquo;knot shadow\u0026rdquo;.\nPolygonal tangles How to form something other than bigons? $6^{\\ast\\ast}$ Minimally polygonal arborescent tangles Conway, J.H. \u0026ldquo;An Enumeration of Knots and Links, and Some of Their Algebraic Properties.\u0026rdquo; In Computational Problems in Abstract Algebra, 329-58. Elsevier, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5\n\u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e Solution? Tabulate the polygonal tangles. Thank you! Special thanks to the committee members: Dr. Isabel Darcy, Supervisor Dr. Francis Bonahon Dr. Keiko Kawamuro Dr. Colleen Mitchell Dr. Radmila Sazdanović Refernces F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html Conway, J.H. \u0026ldquo;An Enumeration of Knots and Links, and Some of Their Algebraic Properties.\u0026rdquo; In Computational Problems in Abstract Algebra, 329-58. Elsevier, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5. Robert Glenn Scharein. Interactive topological drawing. ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D. The University of British Columbia (Canada). URL: https://www.knotplot.com/. Jablan, S., \u0026amp; Sazdanović, R. (2007). Linknot. In Series on Knots and Everything. WORLD SCIENTIFIC. https://doi.org/10.1142/6623 Dowker, C. H., \u0026amp; Thistlethwaite, M. B. (1983). Classification of knot projections. In Topology and its Applications (Vol. 16, Issue 1, pp. 19-31). Elsevier BV. https://doi.org/10.1016/0166-8641(83)90004-4 Hoste, J., Thistlethwaite, M., \u0026amp; Weeks, J. (1998). The first 1,701,936 knots. In The Mathematical Intelligencer (Vol. 20, Issue 4, pp. 33-48). Springer Science and Business Media LLC. https://doi.org/10.1007/bf03025227 Burton, B. A. (2020). The Next 350 Million Knots. Schloss Dagstuhl - Leibniz-Zentrum Für Informatik. https://doi.org/10.4230/LIPICS.SOCG.2020.25 Connolly, Nicholas. Classification and Tabulation of 2-String Tangles: The Astronomy of Subtangle Decompositions. University of Iowa, 2021, https://doi.org/10.17077/etd.005978. Cayley, A. (2009). The collected mathematical papers (Vol. 3). Cambridge University Press. https://doi.org/10.1017/CBO9780511703690 Nakano, S. (2002). Efficient generation of plane trees. In Information Processing Letters (Vol. 84, Issue 3, pp. 167-172). Elsevier BV. https://doi.org/10.1016/s0020-0190(02)00240-5 Refernces Facebook, Public domain, via Wikimedia Commons FastAPI The MIT License (MIT) Carlos Baraza, CC0, via Wikimedia Commons Qq1040058283, Public domain, via Wikimedia Commons Jeremy Kratz, Public domain, via Wikimedia Commons Cython and Python, Apache License 2.0, via Wikimedia Commons mermaidjs www.python.org, GPL, via Wikimedia Commons Mongodb Ryan Dahl, MIT, via Wikimedia Commons Holger Krekel, CC BY 2.5, via Wikimedia Commons Alon Zakai, MIT, via Wikimedia Commons Cmake team. The original uploader was Francesco Betti Sorbelli at Italian Wikipedia.. Vectorized by Magasjukur2, CC BY 2.0, via Wikimedia Commons Emoji One, CC BY-SA 4.0, via Wikimedia Commons NW x Y \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e ","date":"18 August 2025","externalUrl":null,"permalink":"/speaking/research/defense/","section":"Slides","summary":"Talk for my thesis defense on tangle tabulation.","title":"Defense Slides","type":"slides"},{"content":"","date":"18 August 2025","externalUrl":null,"permalink":"/","section":"Joe Starr","summary":"","title":"Joe Starr","type":"page"},{"content":"","date":"18 August 2025","externalUrl":null,"permalink":"/speaking/research/","section":"Slides","summary":"","title":"Research","type":"speaking"},{"content":"","date":"18 August 2025","externalUrl":null,"permalink":"/tags/research-talks/","section":"Tags","summary":"","title":"Research Talks","type":"tags"},{"content":"","date":"18 August 2025","externalUrl":null,"permalink":"/speaking/","section":"Slides","summary":"","title":"Slides","type":"speaking"},{"content":"","date":"18 August 2025","externalUrl":null,"permalink":"/tags/","section":"Tags","summary":"","title":"Tags","type":"tags"},{"content":"","date":"18 August 2025","externalUrl":null,"permalink":"/tags/talks/","section":"Tags","summary":"","title":"Talks","type":"tags"},{"content":"","date":"16 August 2025","externalUrl":null,"permalink":"/tags/index/","section":"Tags","summary":"","title":"Index","type":"tags"},{"content":"","date":"16 August 2025","externalUrl":null,"permalink":"/resources/","section":"Resources","summary":"","title":"Resources","type":"resources"},{"content":"","date":"16 August 2025","externalUrl":null,"permalink":"/resources/tools/","section":"Resources","summary":"","title":"Tools","type":"resources"},{"content":"","date":"6 August 2025","externalUrl":null,"permalink":"/misc/","section":"","summary":"","title":"","type":"misc"},{"content":"Knots and links are much more common in design than people might realize. This page documents knots that I find \u0026ldquo;in the wild\u0026rdquo;, ranging from knots in jewelry to knots on building facads.\nFor each knot I find, when I have time, I\u0026rsquo;ll add some information or commentary about it.\nSubmit a knot! ","date":"6 August 2025","externalUrl":null,"permalink":"/misc/knots_in_the_wild/","section":"","summary":"","title":"Knots in the Wild","type":"misc"},{"content":"","date":"6 August 2025","externalUrl":null,"permalink":"/series/knots-in-the-wild/","section":"Series","summary":"","title":"Knots in the Wild","type":"series"},{"content":"","date":"6 August 2025","externalUrl":null,"permalink":"/series/","section":"Series","summary":"","title":"Series","type":"series"},{"content":"","date":"28 May 2025","externalUrl":null,"permalink":"/resources/general/","section":"Resources","summary":"","title":"General Resources","type":"resources"},{"content":" Preparing undergraduate researchers for graduate school by developing their instruction. The course The course, ISA:4040 Teaching Your Undergraduate Research is based on the prior work of Hoffmann and Lenoch (Hoffmann \u0026amp; Lenoch, 2013) but reimagined for undergraduates engaged in undergraduate research. The course\u0026rsquo;s established goal is to provide experience in instructional design and implementation to undergraduates whose intention is to continue to graduate school. The course has been offered for approximately five years by The Iowa Sciences Academy a division of The University of Iowa, whose mission is to support the success of undergraduate students interested in research and scientific communication. In that time the design of the course has primarily been that of a traditional course, relying on lecture and written/oral assignments taking the form of a term project. The term project asked students to produce a teaching unit for a general audience lesson on the student\u0026rsquo;s own research work. This teaching unit design and execution were graded against a common rubric.\nThe Redesign The redesigned course adopts modern instructional techniques such as a gradual release of responsibility model (Fisher \u0026amp; Frey, 2013) and ungraded assessments (Stommel, 2023).\nThe Design Goals The following goals were established to drive the redesign of the course:\nStudents can engage in teaching as a structured activity. Students are confident to \u0026ldquo;go rogue\u0026rdquo; in their own learning. Students acknowledge that failure breeds success. The Plan: Gradual Release The course began with a model lesson presented by the instructor, based on the instructor\u0026rsquo;s own research. This was followed by a group discussion where the design of the model lesson was unpacked giving a practical example for backward design in the wild. This was supplemented with specific readings from Understanding by Design (Wiggins \u0026amp; McTighe, 2008). Students were given more agency as the course moved to discussion of assessment. In this section of the course, students were given specific sources and empowered to select their own specific readings. The readings were discussed in a structured jigsaw activity where students were guided in sharing their learning from their selected readings. Students were then asked to prepare and present a five-minute lighting version of their lesson. The final section of instructional content was a peer review of drafts of their lesson. Students were asked to structure their own review requests where they prompted peers for specific feedback relevant to their particular needs. Following the main instructional portion of the course each student took fifty minutes each to give their lesson including prompting feedback from their peers. After all students have completed their lesson, each student conducted a qualitative analysis of their peer given feedback. This modeled a complete life cycle for course creation for each student.\nThe Plan: Ungrading The redesigned course has four deliverables for students combined into a portfolio. First, students were still expected to create the teaching unit. Second is a collection of the students\u0026rsquo; design documentation detailing the evolution of the goals and design of their lesson throughout the semester. Students are presented with a backward design and encouraged to utilize it in their design process. As they moved through the course activities they were invited to change their minds, \u0026ldquo;go rogue\u0026rdquo;, and retain the artifacts of their evolving work. Third, is a collection of reflections on supplemental work. These supplemental assignments were designed and selected by the student. These included but was not limited to; literature reviews of teaching research, substantive extensions of their teaching unit, or self-determined mini-projects impacting their development in teaching. The final item in the portfolio is a collection of peer observation prompts and responses written and analyzed by the student. The redesigned course moves away from a unified rubric and adopts a concept of ungrading built from the ideas of Jesse Stommel (Stommel, 2023) as well as discussions with practitioners who have implemented similar methods. The ISA:4040 upgrading strategy fits closely the professional one-to-one and performance review paradigm. Students were required to attend office hour one-to-ones with a minimum once a month plus once on the week before they teach, but were encouraged to attend office hours as frequently as desired. During these meetings the students\u0026rsquo; personal course objectives were discussed, agreed on, and progress evaluated. The semester culminated with a performance review where the portfolio is reviewed, and a course retrospective establishes a grade against the agreed on course objectives.\nThe Research As a portion of a more extensive scholarly review of the Teaching Your Undergraduate Research course paradigm data was collected on experience of students in this course. Data was collected from students (n=7) during interviews which were conducted during final retrospective meetings.\nThe Interview The interview was structured with a standardized question set which addressed the following areas of interest:\nGeneral Experience Teaching Unit Development Learning Outcomes Feedback and Revision Impact of Grading Methods Broader Benefits Preliminary Report Data is still being processed as such only preliminary results are presented here. Student feedback during touchpoints indicates an increase apperception for the intricacies of instruction as well as increased confidence in students' self-identification as an instructor. Ungrading, seems to have allowed for students to decouple the failure of an aspect of their preparation from the students\u0026rsquo; failure in the course.\nOverall, the gradual release and ungrading models were successful for this small group of undergraduates. Extending these ideas to other course sections would provide additional broader data that would help us disaggregate data across several subgroups of students. We will continue using these instructional strategies and look for others that will help support our three goals.\nFisher, D., \u0026amp; Frey, N. (2013). Better Learning Through Structured Teaching: A Framework for the Gradual Release of Responsibility. ASCD.\nHoffmann, D., \u0026amp; Lenoch, S. (2013). Teaching Your Research: A Workshop to Teach Curriculum Design to Graduate Students and Post-doctoral Fellows. Medical Science Educator, 23(3), 336-345.\nStommel, J. (2023). Undoing the Grade: Why We Grade, and How to Stop. Hybrid Pedagogy Inc..\nWiggins, G., \u0026amp; McTighe, J. (2008). Understanding by Design. Association for Supervision and Curriculum Development.\n","date":"27 February 2025","externalUrl":null,"permalink":"/speaking/posters/rume_25/","section":"Slides","summary":"","title":"27th SIGMAA on RUME Conference","type":"betterposter"},{"content":"","date":"27 February 2025","externalUrl":null,"permalink":"/speaking/posters/","section":"Slides","summary":"","title":"Posters","type":"speaking"},{"content":"","date":"27 February 2025","externalUrl":null,"permalink":"/tags/posters/","section":"Tags","summary":"","title":"Posters","type":"tags"},{"content":"","date":"10 February 2025","externalUrl":null,"permalink":"/resources/course_plans/","section":"Resources","summary":"","title":"Course Plans","type":"resources"},{"content":"","date":"7 January 2025","externalUrl":null,"permalink":"/teaching/","section":"Instruction","summary":"","title":"Instruction","type":"teaching"},{"content":"","date":"7 January 2025","externalUrl":null,"permalink":"/tags/ta/","section":"Tags","summary":"","title":"TA","type":"tags"},{"content":" $\\newcommand{\\N}{\\mathbb{N}} \\newcommand{\\Z}{\\mathbb{Z}} \\newcommand{\\Q}{\\mathbb{Q}} \\newcommand{\\R}{\\mathbb{R}} \\newcommand{\\LP}{\\left(} \\newcommand{\\RP}{\\right)} \\newcommand{\\LS}{\\left\\lbrace} \\newcommand{\\RS}{\\right\\rbrace} \\newcommand{\\LA}{\\left\\langle} \\newcommand{\\RA}{\\right\\rangle} \\newcommand{\\LB}{\\left[} \\newcommand{\\RB}{\\right]} \\newcommand{\\MM}{\\ \\middle|\\ } \\newcommand{\\exp}{\\text{exp}} \\newcommand{\\abs}[1]{\\left\\vert#1\\right\\vert} \\newcommand{\\msr}[1]{m\\left(#1\\right)} \\newcommand{\\inv}[1]{#1^{-1}} \\newcommand{\\bkt}[1]{\\LA \\img{#1}\\RA} \\require{color}$ $ $\nThe Tanglenomicon A table of two string tangles The algebraic tangles Zachary Bryhtan, Nicholas Connolly, Isabel Darcy, Paria Karimi, Ethan Rooke, Joseph Starr* Mathematics Department at The University of Iowa Joint Math Meetings 2025 (1/10/25) Tangles \u0026ldquo;We define a tangle as a portion of a knot diagram from which there emerge just 4 arcs pointing in the compass directions NW, NE, SW, SE.\u0026rdquo; - Conway, J.H.\nNWNESWSE Conway, J.H. \u0026ldquo;An Enumeration of Knots and Links, and Some of Their Algebraic Properties.\u0026rdquo; In Computational Problems in Abstract Algebra, 329-58. Elsevier, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5\n$\\quad$ $\\quad$ $\\quad$ Basic Operations Operation $+$ $+$ $=$ $=$ $=$ $2$ Operation $\\vee$ $\\vee$ $=$ $=$ $=$ $\\frac{1}{2}$ Algebraic Tangles All possible tangles made from $+$ and $\\vee$ on basic tangles\nAlgebraic A tangle build from $\\vee$ and $+$ on some rational tangles. $$\\LP\\color{var(--r-Purple)}\\LB3\\ 2\\ 3\\RB+\\LB3\\ 2\\ 3\\RB\\color{var(--r-Foreground)}\\RP\\vee\\LP\\color{var(--r-Purple)}\\LB3\\ 2\\ 3\\RB+\\LB3\\ 2\\ 3\\RB\\color{var(--r-Foreground)}\\RP$$ But actually arborescent Arborescent knots (and tangles) are constructed by taking a collection of twisted bands described by a weighted tree and connecting them with successive plumbing.\nIt\u0026rsquo;s straight forward to see (you should see in the example) that algebraic and arborescent constructions describe the same class of object.\nF. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nx Y x Y \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e -2 3 2 -3 0 4 3 \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e 3 2 -3 0 4 {ι,ξ,ς,η} Anatomy of a tree Rings 2 1 2 -2 -1 -2 $\\ $ F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\n3 2 -3 0 4 3 1 2 2 1 2 2 3 2 -3 2 0 4 3 1 1 2 2 $\\ $ Definition The count of rings attached to a vertex is the Ring Number of the vertex. Ring numbers are noted as an augmentation of the vertex. Definition A vertex with ring number $\\geq 1$ or valence $\\geq 3$ is called an Essential Vertex. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\n3 2 -3 4 3 3 -3 4 3 Definition The subtrees remaining after excising all essential vertices and their bonds (half edges) are called the Sticks of a tree. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nLinearization of Arborescent tangles \u0026#953;((2[3][-3])(((([3][4])[8][2 1])-2)2)) \u0026#953; $\\pm$ Abbreviated Canonical Tangle Trees Weight Condition At each vertex of $\\Gamma$, at most one weight is non-zero. Stick Condition On any stick the weights are non-zero except for end vertices that have a bond free in $\\Gamma_0$ and for the case $\\Gamma_0=\\alpha(0[0])$, $\\Gamma_0=\\alpha[0]$. The non-zero weights along any stick are of alternating sign. No end vertex of a stick has weight $\\pm 1$ unless it has a bond free in $\\Gamma_0$, or $\\Gamma_0=[\\pm 1]$. One of: Positivity Condition There are no sticks in $\\Gamma_0$ of the form $[-1],\\ [ -2],\\ \\alpha[-2],\\ \\alpha[2]\\alpha$. Negative Condition There are no sticks in $\\Gamma_0$ of the form $[1],\\ [ 2],\\ \\alpha[2],\\ \\alpha[2]\\alpha$ Abbreviation Condition A vertex of $\\Gamma_0$ of ring number $1$ has valence $\\geq 2$, and $\\Gamma_0$ is not $\\alpha\\langle 2\\ 0\\rangle$. Every ring subtree of $\\Gamma$ is adjacent to a non-essential vertex. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nTheorem (Bonahon and Siebenmann) Consider two $\\pm$ canonical abbreviated arborescent tangle trees $\\Gamma$ and $\\Gamma^\\prime$. Plumbing according to $\\Gamma$ and $\\Gamma^\\prime$ gives isomorphic arborescent tangles if and only if $\\Gamma$ and $\\Gamma^\\prime$ can be deduced from each other by a sequence of moves in the calculus of arborescent tangle trees (happy to talk about the calculus another time). Note Note $\\pm$ abbreviated canonical tangle trees do not describe minimal diagrams. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nDefinition We call the crossing number of a the tangle diagram described by a $\\pm$ abbreviated canonical tangle tree the canonical arborescent crossing number (CACN) which is given by: $$\\text{CACN}=4\\cdot\\#(\\text{rings in the tree})+\\sum |\\text{weights}|$$ Rational Tangle Trees Montesinos Tangle Trees -3 -3 -3 2 -3 2 \u0026#953; -3 2 -3 2 -3 -3 2 -3 2 Constructing an arborescent tangle Game plan Find a way to generate abstract rooted trees Modify that method to get tangle trees $\\rightarrow$ $\\rightarrow$ Nakano, S. (2002). Efficient generation of plane trees. In Information Processing Letters (Vol. 84, Issue 3, pp. 167-172). Elsevier BV. https://doi.org/10.1016/s0020-0190(02)00240-5\n$\\rightarrow$ $\\rightarrow$ Nakano, S. (2002). Efficient generation of plane trees. In Information Processing Letters (Vol. 84, Issue 3, pp. 167-172). Elsevier BV. https://doi.org/10.1016/s0020-0190(02)00240-5\nBuilding a tangle tree List the integral tangles Find the trees Find the ornaments Hang the ornaments on the trees Example: Building the tangles with canonical arborescent crossing number (CACN) 4 1 -1 2 -2 3 2 -3 -1 -2 1 $\\rightarrow$ Trees with $4$ crossings The integral 4 tangle 4 Call the CACN of a tree $T$ and the CACN of an ornament $O$. For the tangle generated by hanging an ornament to have $\\text{CACN}=4$ we need $T+O=4$.\nAdditionally, hanging an ornament must satisfy the stick condition. Meaning:\n$1\u0026lt;O\\leq 4=\\text{CACN}$ and $0\\leq T\u0026lt;4$ Signs of weights on an ornament may need to be adjusted.$\\text{Example: }[1\\ -2\\ 3]\\to [-1\\ 2\\ -3]$ Tangles with root which is weight $0$ and non-essential are excluded, except $\\iota[0]$. This gives the following for $T$ and $O$: $\\small\\begin{array}{|c|c|} \\hline T \u0026 O \\\\ \\hline 2 \u0026 2 \\\\ \\hline 1 \u0026 3 \\\\ \\hline 0 \u0026 4 \\\\ \\hline \\end{array}$ $\\rightarrow$ $2$ $2$ $\\rightarrow$ $4$ 1 1 2 -2 1 1 2 1 2 1 1 1 2 1 1 2 1 2 1 $\\rightarrow$ -2 1 1 2 -2 2 -2 -2 1 1 2 1 2 1 $\\rightarrow$ $1$ $3$ $\\rightarrow$ $4$ 1 -1 2 2 1 1 2 1 2 1 1 1 2 2 3 -2 1 $\\rightarrow$ -3 1 3 -1 2 -1 1 -2 1 -1 2 2 $\\rightarrow$ $0$ $4$ $\\rightarrow$ $4$ \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e 1 1 2 2 3 3 2 1 2 1 -3 1 2 2 2 -1 1 2 -2 -2 -2 4 $\\rightarrow$ 2 2 -2 2 -2 1 -1 3 -1 -4 2 2 Tangles of canonical arborescent crossing number up to 4 -2 1 -1 -2 -2 -2 -2 1 -3 -2 2 2 -1 2 2 3 2 -2 1 -3 1 -1 3 -1 2 2 -1 1 4 -4 Using The Tanglenomicon Alpha To play with a live version, visit: \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e https://tanglenomicon.com Technologies ThrowTheSwitch/Unity Simple Unit Testing for C C 3.3k 935 joe-starr.com Sources Dror Bar-Natan The Most Important Missing Infrastructure Project in Knot Theory Kauffman, L. H., and S. Lambropoulou. \u0026ldquo;From Tangle Fractions to DNA.\u0026rdquo; In Topology in Molecular Biology, edited by Michail Ilych Monastyrsky, 69-110. Biological and Medical Physics, Biomedical Engineering. Berlin, Heidelberg: Springer Berlin Heidelberg, 2007. https://doi.org/10.1007/978-3-540-49858-2_5. Moon, Hyeyoung, and Isabel K. Darcy. \u0026ldquo;Tangle Equations Involving Montesinos Links.\u0026rdquo; Journal of Knot Theory and Its Ramifications 30, no. 08 (July 2021): 2150060. https://doi.org/10.1142/S0218216521500607. Conway, J.H. \u0026ldquo;An Enumeration of Knots and Links, and Some of Their Algebraic Properties.\u0026rdquo; In Computational Problems in Abstract Algebra, 329-58. Elsevier, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5. Louis H. Kauffman and Sofia Lambropoulou. Classifying and applying rational knots and rational tangles. In DeTurck, editor, Contemporary Mathematics, volume 304, pages 223-259, 2001 Alain Caudron. Classification des nœuds et des enlacements, volume 4 of Publications Math ́ematiques d\u0026rsquo;Orsay 82 [Mathematical Publications of Orsay 82]. Universit ́e de ParisSud, D ́epartement de Mathe ́matique, Orsay, 1982. Robert Glenn Scharein. Interactive topological drawing. ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D. The University of British Columbia (Canada). URL: https://www.knotplot.com/. Jablan, S., \u0026amp; Sazdanović, R. (2007). Linknot. In Series on Knots and Everything. WORLD SCIENTIFIC. https://doi.org/10.1142/6623 Dowker, C. H., \u0026amp; Thistlethwaite, M. B. (1983). Classification of knot projections. In Topology and its Applications (Vol. 16, Issue 1, pp. 19-31). Elsevier BV. https://doi.org/10.1016/0166-8641(83)90004-4 Hoste, J., Thistlethwaite, M., \u0026amp; Weeks, J. (1998). The first 1,701,936 knots. In The Mathematical Intelligencer (Vol. 20, Issue 4, pp. 33-48). Springer Science and Business Media LLC. https://doi.org/10.1007/bf03025227 Burton, B. A. (2020). The Next 350 Million Knots. Schloss Dagstuhl - Leibniz-Zentrum Für Informatik. https://doi.org/10.4230/LIPICS.SOCG.2020.25 C. Livingston and A. H. Moore, KnotInfo: Table of Knot Invariants, knotinfo.math.indiana.edu, today\u0026rsquo;s date (eg. August 24, 2023). Schubert, Horst. \u0026ldquo;Knoten mit zwei Brücken..\u0026rdquo; Mathematische Zeitschrift 65 (1956): 133-170. http://eudml.org/doc/169591. Jos ́e M. Montesinos. Seifert manifolds that are ramified two-sheeted cyclic coverings. Bol. Soc. Mat. Mexicana (2), 18:1-32, 1973. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html Connolly, Nicholas. Classification and Tabulation of 2-String Tangles: The Astronomy of Subtangle Decompositions. University of Iowa, 2021, https://doi.org/10.17077/etd.005978. Nakano, S. (2002). Efficient generation of plane trees. In Information Processing Letters (Vol. 84, Issue 3, pp. 167-172). Elsevier BV. https://doi.org/10.1016/s0020-0190(02)00240-5 Sources Facebook, Public domain, via Wikimedia Commons FastAPI The MIT License (MIT) Carlos Baraza, CC0, via Wikimedia Commons Qq1040058283, Public domain, via Wikimedia Commons Jeremy Kratz, Public domain, via Wikimedia Commons Cython and Python, Apache License 2.0, via Wikimedia Commons mermaidjs www.python.org, GPL, via Wikimedia Commons Mongodb Ryan Dahl, MIT, via Wikimedia Commons Holger Krekel, CC BY 2.5, via Wikimedia Commons Alon Zakai, MIT, via Wikimedia Commons Cmake team. The original uploader was Francesco Betti Sorbelli at Italian Wikipedia.. Vectorized by Magasjukur2, CC BY 2.0, via Wikimedia Commons Emoji One, CC BY-SA 4.0, via Wikimedia Commons Matt Brooks, CC0, via Wikimedia Commons ","date":"4 January 2025","externalUrl":null,"permalink":"/speaking/research/jmm_25/","section":"Slides","summary":"Talk given at the Joint Math Meetings 2025","title":"Joint Math Meetings 2025","type":"slides"},{"content":" $\\newcommand{\\N}{\\mathbb{N}} \\newcommand{\\Z}{\\mathbb{Z}} \\newcommand{\\Q}{\\mathbb{Q}} \\newcommand{\\R}{\\mathbb{R}} \\newcommand{\\LP}{\\left(} \\newcommand{\\RP}{\\right)} \\newcommand{\\LS}{\\left\\lbrace} \\newcommand{\\RS}{\\right\\rbrace} \\newcommand{\\LA}{\\left\\langle} \\newcommand{\\RA}{\\right\\rangle} \\newcommand{\\LB}{\\left[} \\newcommand{\\RB}{\\right]} \\newcommand{\\MM}{\\ \\middle|\\ } \\newcommand{\\exp}{\\text{exp}} \\newcommand{\\abs}[1]{\\left\\vert#1\\right\\vert} \\newcommand{\\msr}[1]{m\\left(#1\\right)} \\newcommand{\\inv}[1]{#1^{-1}} \\newcommand{\\bkt}[1]{\\LA \\img{#1}\\RA} \\require{color}$ $ $\nThe Tanglenomicon A table of two string tangles The algebraic tangles Zachary Bryhtan, Nicholas Connolly, Isabel Darcy, Paria Karimi, Ethan Rooke, Joseph Starr* Mathematics Department at The University of Iowa Knots in Washington (12/7/24) Tangles \u0026ldquo;We define a tangle as a portion of a knot diagram from which there emerge just 4 arcs pointing in the compass directions NW, NE, SW, SE.\u0026rdquo; - Conway, J.H.\nNWNESWSE Conway, J.H. \u0026ldquo;An Enumeration of Knots and Links, and Some of Their Algebraic Properties.\u0026rdquo; In Computational Problems in Abstract Algebra, 329-58. Elsevier, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5\n$\\quad$ $\\quad$ $\\quad$ Basic Operations Operation $+$ $+$ $=$ $=$ $=$ $2$ Operation $\\vee$ $\\vee$ $=$ $=$ $=$ $\\frac{1}{2}$ Algebraic Tangles All possible tangles made from $+$ and $\\vee$ on basic tangles\nAlgebraic A tangle build from $\\vee$ and $+$ on some rational tangles. $$\\LP\\color{var(--r-Purple)}\\LB3\\ 2\\ 3\\RB+\\LB3\\ 2\\ 3\\RB\\color{var(--r-Foreground)}\\RP\\vee\\LP\\color{var(--r-Purple)}\\LB3\\ 2\\ 3\\RB+\\LB3\\ 2\\ 3\\RB\\color{var(--r-Foreground)}\\RP$$ But actually arborescent Arborescent knots (and tangles) are constructed by taking a collection of twisted bands described by a weighted tree and connecting them with successive plumbing.\nIt\u0026rsquo;s straight forward to see (you should see in the example) that algebraic and arborescent constructions describe the same class of object.\nF. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nx Y x Y \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e -2 3 2 -3 0 4 3 \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e 3 2 -3 0 4 {ι,ξ,ς,η} Anatomy of a tree Rings 2 1 2 -2 -1 -2 $\\ $ F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\n3 2 -3 0 4 3 1 2 2 1 2 2 3 2 -3 2 0 4 3 1 1 2 2 $\\ $ Definition The count of rings attached to a vertex is the Ring Number of the vertex. Ring numbers are noted as an augmentation of the vertex. Definition A vertex with ring number $\\geq 1$ or valence $\\geq 3$ is called an Essential Vertex. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\n3 2 -3 4 3 3 -3 4 3 Definition The subtrees remaining after excising all essential vertices and their bonds (half edges) are called the Sticks of a tree. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nLinearization of Arborescent tangles \u0026#953;((2[3][-3])(((([3][4])[8][2 1])-2)2)) \u0026#953; $\\pm$ Abbreviated Canonical Tangle Trees Weight Condition At each vertex of $\\Gamma$, at most one weight is non-zero. Stick Condition On any stick the weights are non-zero except for end vertices that have a bond free in $\\Gamma_0$ and for the case $\\Gamma_0=\\alpha(0[0])$, $\\Gamma_0=\\alpha[0]$. The non-zero weights along any stick are of alternating sign. No end vertex of a stick has weight $\\pm 1$ unless it has a bond free in $\\Gamma_0$, or $\\Gamma_0=[\\pm 1]$. One of: Positivity Condition There are no sticks in $\\Gamma_0$ of the form $[-1],\\ [ -2],\\ \\alpha[-2],\\ \\alpha[2]\\alpha$. Negative Condition There are no sticks in $\\Gamma_0$ of the form $[1],\\ [ 2],\\ \\alpha[2],\\ \\alpha[2]\\alpha$ Abbreviation Condition A vertex of $\\Gamma_0$ of ring number $1$ has valence $\\geq 2$, and $\\Gamma_0$ is not $\\alpha\\langle 2\\ 0\\rangle$. Every ring subtree of $\\Gamma$ is adjacent to a non-essential vertex. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nTheorem (Bonahon and Siebenmann) Consider two $\\pm$ canonical abbreviated arborescent tangle trees $\\Gamma$ and $\\Gamma^\\prime$. Plumbing according to $\\Gamma$ and $\\Gamma^\\prime$ gives isomorphic arborescent tangles if and only if $\\Gamma$ and $\\Gamma^\\prime$ can be deduced from each other by a sequence of moves in the calculus of arborescent tangle trees (happy to talk about the calculus another time). Note Note $\\pm$ abbreviated canonical tangle trees do not describe minimal diagrams. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nDefinition We call the crossing number of a the tangle diagram described by a $\\pm$ abbreviated canonical tangle tree the canonical arborescent crossing number (CACN) which is given by: $$\\text{CACN}=4\\cdot\\#(\\text{rings in the tree})+\\sum |\\text{weights}|$$ Rational Tangle Trees Montesinos Tangle Trees -3 -3 -3 2 -3 2 \u0026#953; -3 2 -3 2 -3 -3 2 -3 2 Constructing an arborescent tangle Game plan Find a way to generate abstract rooted trees Modify that method to get tangle trees $\\rightarrow$ $\\rightarrow$ Nakano, S. (2002). Efficient generation of plane trees. In Information Processing Letters (Vol. 84, Issue 3, pp. 167-172). Elsevier BV. https://doi.org/10.1016/s0020-0190(02)00240-5\n$\\rightarrow$ $\\rightarrow$ Nakano, S. (2002). Efficient generation of plane trees. In Information Processing Letters (Vol. 84, Issue 3, pp. 167-172). Elsevier BV. https://doi.org/10.1016/s0020-0190(02)00240-5\nBuilding a tangle tree List the integral tangles Find the trees Find the ornaments Hang the ornaments on the trees Example: Building the tangles with canonical arborescent crossing number (CACN) 4 1 -1 2 -2 3 2 -3 -1 -2 1 $\\rightarrow$ Trees with $4$ crossings The integral 4 tangle 4 Call the CACN of a tree $T$ and the CACN of an ornament $O$. For the tangle generated by hanging an ornament to have $\\text{CACN}=4$ we need $T+O=4$.\nAdditionally, hanging an ornament must satisfy the stick condition. Meaning:\n$1\u0026lt;O\\leq 4=\\text{CACN}$ and $0\\leq T\u0026lt;4$ Signs of weights on an ornament may need to be adjusted.$\\text{Example: }[1\\ -2\\ 3]\\to [-1\\ 2\\ -3]$ Tangles with root which is weight $0$ and non-essential are excluded, except $\\iota[0]$. This gives the following for $T$ and $O$: $\\small\\begin{array}{|c|c|} \\hline T \u0026 O \\\\ \\hline 2 \u0026 2 \\\\ \\hline 1 \u0026 3 \\\\ \\hline 0 \u0026 4 \\\\ \\hline \\end{array}$ $\\rightarrow$ $2$ $2$ $\\rightarrow$ $4$ 1 1 2 -2 1 1 2 1 2 1 1 1 2 1 1 2 1 2 1 $\\rightarrow$ -2 1 1 2 -2 2 -2 -2 1 1 2 1 2 1 $\\rightarrow$ $1$ $3$ $\\rightarrow$ $4$ 1 -1 2 2 1 1 2 1 2 1 1 1 2 2 3 -2 1 $\\rightarrow$ -3 1 3 -1 2 -1 1 -2 1 -1 2 2 $\\rightarrow$ $0$ $4$ $\\rightarrow$ $4$ \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e 1 1 2 2 3 3 2 1 2 1 -3 1 2 2 2 -1 1 2 -2 -2 -2 4 $\\rightarrow$ 2 2 -2 2 -2 1 -1 3 -1 -4 2 2 Tangles of canonical arborescent crossing number up to 4 -2 1 -1 -2 -2 -2 -2 1 -3 -2 2 2 -1 2 2 3 2 -2 1 -3 1 -1 3 -1 2 2 -1 1 4 -4 Using The Tanglenomicon Alpha To play with a live version, visit: \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e https://tanglenomicon.com Technologies ThrowTheSwitch/Unity Simple Unit Testing for C C 3.3k 935 joe-starr.com Sources Dror Bar-Natan The Most Important Missing Infrastructure Project in Knot Theory Kauffman, L. H., and S. Lambropoulou. \u0026ldquo;From Tangle Fractions to DNA.\u0026rdquo; In Topology in Molecular Biology, edited by Michail Ilych Monastyrsky, 69-110. Biological and Medical Physics, Biomedical Engineering. Berlin, Heidelberg: Springer Berlin Heidelberg, 2007. https://doi.org/10.1007/978-3-540-49858-2_5. Moon, Hyeyoung, and Isabel K. Darcy. \u0026ldquo;Tangle Equations Involving Montesinos Links.\u0026rdquo; Journal of Knot Theory and Its Ramifications 30, no. 08 (July 2021): 2150060. https://doi.org/10.1142/S0218216521500607. Conway, J.H. \u0026ldquo;An Enumeration of Knots and Links, and Some of Their Algebraic Properties.\u0026rdquo; In Computational Problems in Abstract Algebra, 329-58. Elsevier, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5. Louis H. Kauffman and Sofia Lambropoulou. Classifying and applying rational knots and rational tangles. In DeTurck, editor, Contemporary Mathematics, volume 304, pages 223-259, 2001 Alain Caudron. Classification des nœuds et des enlacements, volume 4 of Publications Math ́ematiques d\u0026rsquo;Orsay 82 [Mathematical Publications of Orsay 82]. Universit ́e de ParisSud, D ́epartement de Mathe ́matique, Orsay, 1982. Robert Glenn Scharein. Interactive topological drawing. ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D. The University of British Columbia (Canada). URL: https://www.knotplot.com/. Jablan, S., \u0026amp; Sazdanović, R. (2007). Linknot. In Series on Knots and Everything. WORLD SCIENTIFIC. https://doi.org/10.1142/6623 Dowker, C. H., \u0026amp; Thistlethwaite, M. B. (1983). Classification of knot projections. In Topology and its Applications (Vol. 16, Issue 1, pp. 19-31). Elsevier BV. https://doi.org/10.1016/0166-8641(83)90004-4 Hoste, J., Thistlethwaite, M., \u0026amp; Weeks, J. (1998). The first 1,701,936 knots. In The Mathematical Intelligencer (Vol. 20, Issue 4, pp. 33-48). Springer Science and Business Media LLC. https://doi.org/10.1007/bf03025227 Burton, B. A. (2020). The Next 350 Million Knots. Schloss Dagstuhl - Leibniz-Zentrum Für Informatik. https://doi.org/10.4230/LIPICS.SOCG.2020.25 C. Livingston and A. H. Moore, KnotInfo: Table of Knot Invariants, knotinfo.math.indiana.edu, today\u0026rsquo;s date (eg. August 24, 2023). Schubert, Horst. \u0026ldquo;Knoten mit zwei Brücken..\u0026rdquo; Mathematische Zeitschrift 65 (1956): 133-170. http://eudml.org/doc/169591. Jos ́e M. Montesinos. Seifert manifolds that are ramified two-sheeted cyclic coverings. Bol. Soc. Mat. Mexicana (2), 18:1-32, 1973. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html Connolly, Nicholas. Classification and Tabulation of 2-String Tangles: The Astronomy of Subtangle Decompositions. University of Iowa, 2021, https://doi.org/10.17077/etd.005978. Nakano, S. (2002). Efficient generation of plane trees. In Information Processing Letters (Vol. 84, Issue 3, pp. 167-172). Elsevier BV. https://doi.org/10.1016/s0020-0190(02)00240-5 Sources Facebook, Public domain, via Wikimedia Commons FastAPI The MIT License (MIT) Carlos Baraza, CC0, via Wikimedia Commons Qq1040058283, Public domain, via Wikimedia Commons Jeremy Kratz, Public domain, via Wikimedia Commons Cython and Python, Apache License 2.0, via Wikimedia Commons mermaidjs www.python.org, GPL, via Wikimedia Commons Mongodb Ryan Dahl, MIT, via Wikimedia Commons Holger Krekel, CC BY 2.5, via Wikimedia Commons Alon Zakai, MIT, via Wikimedia Commons Cmake team. The original uploader was Francesco Betti Sorbelli at Italian Wikipedia.. Vectorized by Magasjukur2, CC BY 2.0, via Wikimedia Commons Emoji One, CC BY-SA 4.0, via Wikimedia Commons Matt Brooks, CC0, via Wikimedia Commons ","date":"7 December 2024","externalUrl":null,"permalink":"/speaking/research/kiw50/","section":"Slides","summary":"Talk given at the Knots in Washington 50 conference","title":"Knots in Washington 50","type":"slides"},{"content":" $\\newcommand{\\N}{\\mathbb{N}} \\newcommand{\\Z}{\\mathbb{Z}} \\newcommand{\\Q}{\\mathbb{Q}} \\newcommand{\\R}{\\mathbb{R}} \\newcommand{\\LP}{\\left(} \\newcommand{\\RP}{\\right)} \\newcommand{\\LS}{\\left\\lbrace} \\newcommand{\\RS}{\\right\\rbrace} \\newcommand{\\LA}{\\left\\langle} \\newcommand{\\RA}{\\right\\rangle} \\newcommand{\\LB}{\\left[} \\newcommand{\\RB}{\\right]} \\newcommand{\\MM}{\\ \\middle|\\ } \\newcommand{\\exp}{\\text{exp}} \\newcommand{\\abs}[1]{\\left\\vert#1\\right\\vert} \\newcommand{\\msr}[1]{m\\left(#1\\right)} \\newcommand{\\inv}[1]{#1^{-1}} \\newcommand{\\bkt}[1]{\\LA \\img{#1}\\RA} \\require{color}$ $\\newcommand{mvspc}{\\hspace{.125cm}}$ $\\newcommand{mvtwt}{\\frac{\\mvspc}{\\mvspc}\\frac{\\mvspc}{\\mvspc}}$ $\\newcommand{mvtht}{\\frac{\\mvspc}{\\mvspc}\\frac{\\mvspc}{\\mvspc}\\frac{\\mvspc}{\\mvspc}}$ Arborescent knots (and tangles) are constructed by taking a collection of twisted bands described by a weighted tree and connecting them with successive plumbing.\nF. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nx Y x Y \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e -2 W0 W2 W1 C0 C2 C1 3 2 -3 0 4 3 \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e 3 2 -3 0 4 {ι,ξ,ς,η} Anatomy of a tree Rings 2 1 2 -2 -1 -2 $\\ $ F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\n3 2 -3 0 4 3 1 2 2 1 2 2 3 2 -3 2 0 4 3 1 1 2 2 $\\ $ Definition The count of rings attached to a vertex is the Ring Number of the vertex. Ring numbers are noted as an augmentation of the vertex. Definition A vertex with ring number $\\geq 1$ or valence $\\geq 3$ is called an Essential Vertex. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\n3 2 -3 4 3 3 -3 4 3 Definition The subtrees remaining after excising all essential vertices and their bonds (half edges) are called the Sticks of a tree. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nLinearization of Weighted Arborescent Tangle Trees \u0026#953;((2[3][-3])(((([3][4])[8][2 1])-2)2)) \u0026#953; W0 W2 W1 C0 C2 C1 Linearizing a vertex locally We imagine an arm sweeping out from the lowest index bond (parent) \u0026ldquo;picking up\u0026rdquo; the data of each weight and child as it sweeps across the data.\nLinearizing a tree To linearize an entire tree, we start at the root and linearize each vertex. However, when we sweep over a child, we descend to that child and linearize. When we have completed linearizing a vertex we pop up the tree.\nDelimiting depth As we move up and down the tree, we need to keep track how deep we are into the tree. When we descend we add an open delimiter. When we ascend, we add a closing delimiter. The delimiters we will use are:\n$\\LB\\ \\ \\RB$: Corresponds to a half open stick and is interpreted as a twist vector for a rational tangle. Note that, to align with the traditional notation, the twist vector is written in depth first post order. $\\LP\\ \\ \\RP$: Corresponds to a vertex not on a half open stick with no ring number. $\\LA\\ \\ \\RA$: Corresponds to a vertex not on a half open stick with ring number. \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e Sources Dror Bar-Natan The Most Important Missing Infrastructure Project in Knot Theory Kauffman, L. H., and S. Lambropoulou. \u0026ldquo;From Tangle Fractions to DNA.\u0026rdquo; In Topology in Molecular Biology, edited by Michail Ilych Monastyrsky, 69-110. Biological and Medical Physics, Biomedical Engineering. Berlin, Heidelberg: Springer Berlin Heidelberg, 2007. https://doi.org/10.1007/978-3-540-49858-2_5. Moon, Hyeyoung, and Isabel K. Darcy. \u0026ldquo;Tangle Equations Involving Montesinos Links.\u0026rdquo; Journal of Knot Theory and Its Ramifications 30, no. 08 (July 2021): 2150060. https://doi.org/10.1142/S0218216521500607. Conway, J.H. \u0026ldquo;An Enumeration of Knots and Links, and Some of Their Algebraic Properties.\u0026rdquo; In Computational Problems in Abstract Algebra, 329-58. Elsevier, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5. Louis H. Kauffman and Sofia Lambropoulou. Classifying and applying rational knots and rational tangles. In DeTurck, editor, Contemporary Mathematics, volume 304, pages 223-259, 2001 Alain Caudron. Classification des nœuds et des enlacements, volume 4 of Publications Math ́ematiques d\u0026rsquo;Orsay 82 [Mathematical Publications of Orsay 82]. Universit ́e de ParisSud, D ́epartement de Mathe ́matique, Orsay, 1982. Robert Glenn Scharein. Interactive topological drawing. ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D. The University of British Columbia (Canada). URL: https://www.knotplot.com/. Jablan, S., \u0026amp; Sazdanović, R. (2007). Linknot. In Series on Knots and Everything. WORLD SCIENTIFIC. https://doi.org/10.1142/6623 Dowker, C. H., \u0026amp; Thistlethwaite, M. B. (1983). Classification of knot projections. In Topology and its Applications (Vol. 16, Issue 1, pp. 19-31). Elsevier BV. https://doi.org/10.1016/0166-8641(83)90004-4 Hoste, J., Thistlethwaite, M., \u0026amp; Weeks, J. (1998). The first 1,701,936 knots. In The Mathematical Intelligencer (Vol. 20, Issue 4, pp. 33-48). Springer Science and Business Media LLC. https://doi.org/10.1007/bf03025227 Burton, B. A. (2020). The Next 350 Million Knots. Schloss Dagstuhl - Leibniz-Zentrum Für Informatik. https://doi.org/10.4230/LIPICS.SOCG.2020.25 C. Livingston and A. H. Moore, KnotInfo: Table of Knot Invariants, knotinfo.math.indiana.edu, today\u0026rsquo;s date (eg. August 24, 2023). Schubert, Horst. \u0026ldquo;Knoten mit zwei Brücken..\u0026rdquo; Mathematische Zeitschrift 65 (1956): 133-170. http://eudml.org/doc/169591. Jos ́e M. Montesinos. Seifert manifolds that are ramified two-sheeted cyclic coverings. Bol. Soc. Mat. Mexicana (2), 18:1-32, 1973. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html Connolly, Nicholas. Classification and Tabulation of 2-String Tangles: The Astronomy of Subtangle Decompositions. University of Iowa, 2021, https://doi.org/10.17077/etd.005978. Nakano, S. (2002). Efficient generation of plane trees. In Information Processing Letters (Vol. 84, Issue 3, pp. 167-172). Elsevier BV. https://doi.org/10.1016/s0020-0190(02)00240-5 Sources Facebook, Public domain, via Wikimedia Commons FastAPI The MIT License (MIT) Carlos Baraza, CC0, via Wikimedia Commons Qq1040058283, Public domain, via Wikimedia Commons Jeremy Kratz, Public domain, via Wikimedia Commons Cython and Python, Apache License 2.0, via Wikimedia Commons mermaidjs www.python.org, GPL, via Wikimedia Commons Mongodb Ryan Dahl, MIT, via Wikimedia Commons Holger Krekel, CC BY 2.5, via Wikimedia Commons Alon Zakai, MIT, via Wikimedia Commons Cmake team. The original uploader was Francesco Betti Sorbelli at Italian Wikipedia.. Vectorized by Magasjukur2, CC BY 2.0, via Wikimedia Commons Emoji One, CC BY-SA 4.0, via Wikimedia Commons Matt Brooks, CC0, via Wikimedia Commons ","date":"7 December 2024","externalUrl":null,"permalink":"/speaking/misc/linearization/","section":"Slides","summary":"","title":"Linearization","type":"slides"},{"content":"","date":"7 December 2024","externalUrl":null,"permalink":"/speaking/misc/","section":"Slides","summary":"","title":"misc","type":"speaking"},{"content":" $\\newcommand{\\N}{\\mathbb{N}} \\newcommand{\\Z}{\\mathbb{Z}} \\newcommand{\\Q}{\\mathbb{Q}} \\newcommand{\\R}{\\mathbb{R}} \\newcommand{\\LP}{\\left(} \\newcommand{\\RP}{\\right)} \\newcommand{\\LS}{\\left\\lbrace} \\newcommand{\\RS}{\\right\\rbrace} \\newcommand{\\LA}{\\left\\langle} \\newcommand{\\RA}{\\right\\rangle} \\newcommand{\\LB}{\\left[} \\newcommand{\\RB}{\\right]} \\newcommand{\\MM}{\\ \\middle|\\ } \\newcommand{\\exp}{\\text{exp}} \\newcommand{\\abs}[1]{\\left\\vert#1\\right\\vert} \\newcommand{\\msr}[1]{m\\left(#1\\right)} \\newcommand{\\inv}[1]{#1^{-1}} \\newcommand{\\bkt}[1]{\\LA \\img{#1}\\RA} \\require{color}$ The Tanglenomicon Zachary Bryhtan, Nicholas Connolly, Isabel Darcy, Ethan Rooke, Joseph Starr* Mathematics Department at The University of Iowa Knots $\\quad$ $\\quad$ $\\quad$ https://www.knotplot.com/\nThe natural question How many knots? Knot Tables Lord Kelvin\u0026rsquo;s vortex theory of the atom Atoms are knotted vortices in the æther. By Hand 1860\u0026rsquo;s Tait computes knots up to 7 crossings 15 knots 1870\u0026rsquo;s Tait, Kirkman, and Little compute knots up to 10 crossings Takes about 25 years 250 knots 1960\u0026rsquo;s Conway computes knots up to 11 crossings \u0026ldquo;A few hours\u0026rdquo; 802 knots By Computer 1980\u0026rsquo;s Dowker and Thistlethwaite compute up to 13 crossings First using a computer 12,966 knots 1990\u0026rsquo;s Hoste, Thistlethwaite, and Weeks compute up to 16 crossings Computer runtime on the order of weeks 1,701,936 knots 2020\u0026rsquo;s Burton computes up to 19 crossings 350 Million knots Conway How did Conway compute 25 years of work in \"a few hours\"? Tangles \u0026ldquo;We define a tangle as a portion of a knot diagram from which there emerge just 4 arcs pointing in the compass directions NW, NE, SW, SE.\u0026rdquo; - Conway, J.H.\nConway, J.H. \u0026ldquo;An Enumeration of Knots and Links, and Some of Their Algebraic Properties.\u0026rdquo; In Computational Problems in Abstract Algebra, 329-58. Elsevier, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5\n$\\quad$ $\\quad$ NWNESWSE $\\quad$ $\\quad$ $\\quad$ Basic Operations Operation $+$ $+$ $=$ $=$ $=$ $2$ Operation $\\vee$ $\\vee$ $=$ $=$ $=$ $\\frac{1}{2}$ The Tanglenomicon A table of two string tangles (up to fixed boundary) Building up $\\ $ $\\ $ $\\ $ $\\ $ Where we are Rational Tangles 8,388,608 up to 23 crossings $\\ $ $\\begin{aligned}\\to\u0026\\ \\LP 3 \\vee \\frac{1}{2}\\RP + 2\\\\\u0026\\\\ \\to\u0026\\ [3\\ 2\\ 2]\\end{aligned}$ Montesinos 120,344,744 up to 23 crossings with non-fixed boundary $+$ $=$ $$\\ =\\ $$ $$=[3\\ 2\\ 0] + [3\\ 2\\ 0]$$ Generalized Montesinos Operation $\\circ$ $\\ $ $= \\color{var(--r-Purple)}([1\\ 2\\ 0] + [1\\ 2\\ 0] + [1\\ 1\\ 0]) \\color{var(--r-Foreground)}\\circ \\color{var(--r-Red)}[2\\ 2]$ Moon, Hyeyoung, and Isabel K. Darcy. \u0026ldquo;Tangle Equations Involving Montesinos Links.\u0026rdquo; Journal of Knot Theory and Its Ramifications 30, no. 08 (July 2021): 2150060. https://doi.org/10.1142/S0218216521500607.\nWhere we\u0026rsquo;re going Algebraic (Arborescent) All possible tangles made from $+$ and $\\vee$ on basic tangles\nAlgebraic A tangle build from $\\vee$ and $+$ on some rational tangles. $$\\LP\\color{var(--r-Purple)}\\LB3\\ 2\\ 3\\RB+\\LB3\\ 2\\ 3\\RB\\color{var(--r-Foreground)}\\RP\\vee\\LP\\color{var(--r-Purple)}\\LB3\\ 2\\ 3\\RB+\\LB3\\ 2\\ 3\\RB\\color{var(--r-Foreground)}\\RP$$ Arborescent Tangles are constructed by taking a collection of twisted bands described by a weighted tree and connecting them with successive plumbing.\nF. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nx Y x Y 3 2 -3 0 4 3 Into the future Non-algebraic/Polygonal 4-valent planar graphs $\\quad$ 4-valent planar graph insertions $6^*\\ *.[1\\ 2\\ 2\\ 3\\ 1].[1\\ 2\\ 2\\ 3\\ 1].[1\\ 2\\ 2\\ 3\\ 1].[1\\ 2\\ 2\\ 3\\ 1].[1\\ 2\\ 2\\ 3\\ 1]$ Technologies ThrowTheSwitch/Unity Simple Unit Testing for C C 3.3k 935 joe-starr.com Sources Dror Bar-Natan The Most Important Missing Infrastructure Project in Knot Theory Kauffman, L. H., and S. Lambropoulou. \u0026ldquo;From Tangle Fractions to DNA.\u0026rdquo; In Topology in Molecular Biology, edited by Michail Ilych Monastyrsky, 69-110. Biological and Medical Physics, Biomedical Engineering. Berlin, Heidelberg: Springer Berlin Heidelberg, 2007. https://doi.org/10.1007/978-3-540-49858-2_5. Moon, Hyeyoung, and Isabel K. Darcy. \u0026ldquo;Tangle Equations Involving Montesinos Links.\u0026rdquo; Journal of Knot Theory and Its Ramifications 30, no. 08 (July 2021): 2150060. https://doi.org/10.1142/S0218216521500607. Conway, J.H. \u0026ldquo;An Enumeration of Knots and Links, and Some of Their Algebraic Properties.\u0026rdquo; In Computational Problems in Abstract Algebra, 329-58. Elsevier, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5. Louis H. Kauffman and Sofia Lambropoulou. Classifying and applying rational knots and rational tangles. In DeTurck, editor, Contemporary Mathematics, volume 304, pages 223-259, 2001 Alain Caudron. Classification des nœuds et des enlacements, volume 4 of Publications Math ́ematiques d\u0026rsquo;Orsay 82 [Mathematical Publications of Orsay 82]. Universit ́e de ParisSud, D ́epartement de Mathe ́matique, Orsay, 1982. Robert Glenn Scharein. Interactive topological drawing. ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D. The University of British Columbia (Canada). URL: https://www.knotplot.com/. Jablan, S., \u0026amp; Sazdanović, R. (2007). Linknot. In Series on Knots and Everything. WORLD SCIENTIFIC. https://doi.org/10.1142/6623 Dowker, C. H., \u0026amp; Thistlethwaite, M. B. (1983). Classification of knot projections. In Topology and its Applications (Vol. 16, Issue 1, pp. 19-31). Elsevier BV. https://doi.org/10.1016/0166-8641(83)90004-4 Hoste, J., Thistlethwaite, M., \u0026amp; Weeks, J. (1998). The first 1,701,936 knots. In The Mathematical Intelligencer (Vol. 20, Issue 4, pp. 33-48). Springer Science and Business Media LLC. https://doi.org/10.1007/bf03025227 Burton, B. A. (2020). The Next 350 Million Knots. Schloss Dagstuhl - Leibniz-Zentrum Für Informatik. https://doi.org/10.4230/LIPICS.SOCG.2020.25 C. Livingston and A. H. Moore, KnotInfo: Table of Knot Invariants, knotinfo.math.indiana.edu, today\u0026rsquo;s date (eg. August 24, 2023). Schubert, Horst. \u0026ldquo;Knoten mit zwei Brücken..\u0026rdquo; Mathematische Zeitschrift 65 (1956): 133-170. http://eudml.org/doc/169591. Jos ́e M. Montesinos. Seifert manifolds that are ramified two-sheeted cyclic coverings. Bol. Soc. Mat. Mexicana (2), 18:1-32, 1973. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html Connolly, Nicholas. Classification and Tabulation of 2-String Tangles: The Astronomy of Subtangle Decompositions. University of Iowa, 2021, https://doi.org/10.17077/etd.005978. Sources Facebook, Public domain, via Wikimedia Commons FastAPI The MIT License (MIT) Carlos Baraza, CC0, via Wikimedia Commons Qq1040058283, Public domain, via Wikimedia Commons Jeremy Kratz, Public domain, via Wikimedia Commons Cython and Python, Apache License 2.0, via Wikimedia Commons mermaidjs www.python.org, GPL, via Wikimedia Commons Mongodb Ryan Dahl, MIT, via Wikimedia Commons Holger Krekel, CC BY 2.5, via Wikimedia Commons Alon Zakai, MIT, via Wikimedia Commons Cmake team. The original uploader was Francesco Betti Sorbelli at Italian Wikipedia.. Vectorized by Magasjukur2, CC BY 2.0, via Wikimedia Commons ","date":"7 September 2024","externalUrl":null,"permalink":"/speaking/research/mathday24/","section":"Slides","summary":"Talk given at the Fall 2024 University of Iowa Math Day","title":"University of Iowa Fall 2024 Math Day","type":"slides"},{"content":" $\\newcommand{\\N}{\\mathbb{N}} \\newcommand{\\Z}{\\mathbb{Z}} \\newcommand{\\Q}{\\mathbb{Q}} \\newcommand{\\R}{\\mathbb{R}} \\newcommand{\\LP}{\\left(} \\newcommand{\\RP}{\\right)} \\newcommand{\\LS}{\\left\\lbrace} \\newcommand{\\RS}{\\right\\rbrace} \\newcommand{\\LA}{\\left\\langle} \\newcommand{\\RA}{\\right\\rangle} \\newcommand{\\LB}{\\left[} \\newcommand{\\RB}{\\right]} \\newcommand{\\MM}{\\ \\middle|\\ } \\newcommand{\\exp}{\\text{exp}} \\newcommand{\\abs}[1]{\\left\\vert#1\\right\\vert} \\newcommand{\\msr}[1]{m\\left(#1\\right)} \\newcommand{\\inv}[1]{#1^{-1}} \\newcommand{\\bkt}[1]{\\LA \\img{#1}\\RA} \\require{color}$ Comprehensive Exam (8/19/24) The Tanglenomicon Zachary Bryhtan, Nicholas Connolly, Isabel Darcy, Ethan Rooke, Joseph Starr* Mathematics Department at The University of Iowa Knots $\\quad$ $\\quad$ $\\quad$ https://www.knotplot.com/\nThe natural question How many knots? Knot Tables Lord Kelvin\u0026rsquo;s vortex theory of the atom Atoms are knotted vortices in the æther. By Hand 1860\u0026rsquo;s Tait computes knots up to 7 crossings 15 knots 1870\u0026rsquo;s Tait, Kirkman, and Little compute knots up to 10 crossings Takes about 25 years 250 knots 1960\u0026rsquo;s Conway computes knots up to 11 crossings \u0026ldquo;A few hours\u0026rdquo; 802 knots By Computer 1980\u0026rsquo;s Dowker and Thistlethwaite compute up to 13 crossings First using a computer 12,966 knots 1990\u0026rsquo;s Hoste, Thistlethwaite, and Weeks compute up to 16 crossings Computer runtime on the order of weeks 1,701,936 knots 2020\u0026rsquo;s Burton computes up to 19 crossings 350 Million knots KnotInfo C. Livingston and A. H. Moore, KnotInfo: Table of Knot Invariants, https://knotinfo.math.indiana.edu/, today\u0026rsquo;s date (eg. August 24, 2023)\nConway How did Conway compute 25 years of work in \"a few hours\"? Tangles \u0026ldquo;We define a tangle as a portion of a knot diagram from which there emerge just 4 arcs pointing in the compass directions NW, NE, SW, SE.\u0026rdquo; - Conway, J.H.\nConway, J.H. \u0026ldquo;An Enumeration of Knots and Links, and Some of Their Algebraic Properties.\u0026rdquo; In Computational Problems in Abstract Algebra, 329-58. Elsevier, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5\n$\\quad$ $\\quad$ NWNESWSE $\\quad$ $\\quad$ $\\quad$ Basic Operations Operation $+$ $+$ $=$ $=$ $=$ $2$ Operation $\\vee$ $\\vee$ $=$ $=$ $=$ $\\frac{1}{2}$ The Tanglenomicon A table of two string tangles (up to fixed boundary) Building up $\\ $ $\\ $ $\\ $ $\\ $ Where we are Rational Tangles $\\ $ $\\begin{aligned}\\to\u0026\\ \\LP 3 \\vee \\frac{1}{2}\\RP + 2\\\\\u0026\\\\ \\to\u0026\\ [3\\ 2\\ 2]\\end{aligned}$ $\\begin{aligned}\\to\u0026\\ \\LP 1 \\vee \\frac{1}{3}\\RP + 1\\\\\u0026\\\\ \\to\u0026\\ [1\\ 3\\ 1]\\end{aligned}$ $\\ $ $\\begin{aligned}\\to\u0026 \\frac{1}{4} + 1\\,\\,\\\\\u0026\\\\ \\to\u0026\\ [4\\ 1]\\end{aligned}$ Generation For any $N$ an obvious twist vector is the twist vector of all $1$s $$[1\\ 1\\ 1\\ \\cdots\\ 1]$$ Noting that when we write this sequence, we have $N-1$ spaces.\nIf we choose to place a $+$ instead of the left most space we get $$[1+1\\ 1\\ \\cdots\\ 1]=[2\\ 1\\ \\cdots\\ 1]$$ we\u0026rsquo;re free to make this choice for each space\nthis gives $N-1$ choices between \u0026lsquo;$+$\u0026rsquo; and space $$[1\\square 1\\square 1\\square\\cdots\\square1]$$ letting us generate twist vectors by simply counting from $0\\to 2^{N-1}$.\nTwist Vectors for $N=5$ $$\\begin{array}{|l|l|l|l|} \\hline [1\\ 1\\ 1\\ 1\\ 1]\\ \u0026\\ [1\\ 1\\ 1\\ 2]\\ \u0026\\ [1\\ 1\\ 2\\ 1]\\ \u0026\\ [1\\ 1\\ 3]\\\\\\hline [1\\ 2\\ 1\\ 1]\\ \u0026\\ [1\\ 2\\ 2]\\ \u0026\\ [1\\ 3\\ 1]\\ \u0026\\ [1\\ 4]\\\\\\hline [2\\ 1\\ 1\\ 1]\\ \u0026\\ [2\\ 1\\ 2]\\ \u0026\\ [2\\ 2\\ 1]\\ \u0026\\ [2\\ 3]\\\\\\hline [3\\ 1\\ 1]\\ \u0026\\ [3\\ 2]\\ \u0026\\ [4\\ 1]\\ \u0026\\ [5]\\\\\\hline \\end{array}$$ Programmatic Description stateDiagram-v2 direction LR state if_done \u0026lt;\u0026lt;choice\u0026gt;\u0026gt; State_i: i=0 State_ipp: i\u0026#43;\u0026#43; state \u0026#34;Construct TV from i as a bitfield\u0026#34; as tv_calc{ state \u0026#34;tmplt=i;j=0;cnt=N\u0026#34; as State_temp State_jpp: j\u0026#43;\u0026#43; State_cntmm: cnt-- State_sum_tv: TV[j]\u0026#43;\u0026#43; State_rsh: tmplt=tmplt\u0026gt;\u0026gt;1 state if_lsb \u0026lt;\u0026lt;choice\u0026gt;\u0026gt; state if_cnteo \u0026lt;\u0026lt;choice\u0026gt;\u0026gt; State_store_tv: Store TV [*] --\u0026gt; State_temp State_temp --\u0026gt; if_cnteo if_cnteo--\u0026gt; State_cntmm: if cnt\u0026gt;0 if_cnteo--\u0026gt; State_store_tv: if cnt==0 State_store_tv --\u0026gt; [*] State_cntmm --\u0026gt;if_lsb if_lsb --\u0026gt;State_sum_tv: if (tmplt \u0026amp; 0x01u)==1u State_sum_tv --\u0026gt; State_rsh if_lsb --\u0026gt;State_jpp: if (tmplt \u0026amp; 0x01u)==0u State_jpp --\u0026gt; State_rsh State_rsh --\u0026gt; if_cnteo } [*] --\u0026gt; State_i State_i --\u0026gt; if_done if_done --\u0026gt; tv_calc: if i \u0026lt; 2**(N-1) tv_calc --\u0026gt; State_ipp State_ipp --\u0026gt; if_done if_done --\u0026gt; [*]: if i == 2**(N-1) i tmplt cnt j tv 0 0000 5 0 [1,1,1,1,1] 0 0000 4 1 [1,1,1,1,1] 0 0000 3 2 [1,1,1,1,1] 0 0000 2 3 [1,1,1,1,1] 0 0000 1 4 [1,1,1,1,1] 0 0000 0 4 [1,1,1,1,1] i tmplt cnt j tv 6 0110 5 4 [1,1,1,1,1] 6 0011 4 1 [1,1,1,1,1] 6 0001 3 1 [1,2,1,1,1] 6 0000 2 1 [1,3,1,1,1] 6 0000 1 2 [1,3,1,1,1] 6 0000 0 2 [1,3,1,1,1] i tmplt cnt j tv 7 0111 5 0 [1,1,1,1,1] 7 0011 4 0 [2,1,1,1,1] 7 0001 3 0 [3,1,1,1,1] 7 0000 2 0 [4,1,1,1,1] 7 0000 1 1 [4,1,1,1,1] 7 0000 0 1 [4,1,1,1,1] Canonical Twist Vectors We can write a canonical twist vector by taking the odd length vectors (appending $0$ where needed).\nCanonical Twist Vectors for $N=5$ $$\\begin{array}{|l|l|l|l|} \\hline [1\\ 1\\ 1\\ 1\\ 1]\\ \u0026\\ [1\\ 1\\ 1\\ 2\\ 0]\\ \u0026\\ [1\\ 1\\ 2\\ 1\\ 0]\\ \u0026\\ [1\\ 1\\ 3]\\\\\\hline [1\\ 2\\ 1\\ 1\\ 0]\\ \u0026\\ [1\\ 2\\ 2]\\ \u0026\\ [1\\ 3\\ 1]\\ \u0026\\ [1\\ 4\\ 0]\\\\\\hline [2\\ 1\\ 1\\ 1\\ 0]\\ \u0026\\ [2\\ 1\\ 2]\\ \u0026\\ [2\\ 2\\ 1]\\ \u0026\\ [2\\ 3\\ 0]\\\\\\hline [3\\ 1\\ 1]\\ \u0026\\ [3\\ 2\\ 0]\\ \u0026\\ [4\\ 1\\ 0]\\ \u0026\\ [5]\\\\\\hline \\end{array}$$ Computations Rational Number (continued fraction) The rational number for a twist vector is computed by taking the twist vector as a finite continued fraction, that is: $$\\LB a\\ b\\ c\\RB=c+\\frac{1}{b+\\frac{1}{a}}$$\nTwist Vector to rational number $$\\ =\\LB 3\\ 2\\ 2\\RB=2+\\frac{1}{2+\\frac{1}{3}}=\\frac{17}{7}$$ J.R. Goldman, L.H. Kauffman, Rational Tangles, Advances in Applied Math., 18 (1997), 300-332. Conway, J.H. \u0026ldquo;An Enumeration of Knots and Links, and Some of Their Algebraic Properties.\u0026rdquo; In Computational Problems in Abstract Algebra, 329-58. Elsevier, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5 To play with twist vectors and continued fractions, visit:\n\u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e https://joe-starr.com/resources/cont_frac_convert/\nParity NWSWSENE NW SW SE NE NWSWSENE NW SW SE NE NWSWSENE NW SW SE NE Computing Parity If we take the rational number $\\frac{p}{q}$ associated with the rational tangle we get the following correspondence for parity\nParity Table $\\begin{array}{|c|c|c|c|} \\hline p\\ \\%\\ 2 \u0026q\\ \\%\\ 2\u0026\\text{Parity}\\\\ \\hline 0 \u00260\u0026N/A\u0026\\\\ \\hline 0 \u00261\u0026 0 \u0026 \\img{/presentations/comp/0.svg}\\\\ \\hline 1 \u00260\u0026\\infty\u0026 \\img{/presentations/comp/inf.svg}\\\\ \\hline 1 \u00261\u0026 1\u0026 \\img{/presentations/comp/parity_1.svg}\\\\ \\hline \\end{array}$ Louis H. Kauffman and Sofia Lambropoulou. Classifying and applying rational knots and rational tangles. In DeTurck, editor, Contemporary Mathematics, volume 304, pages 223-259, 2001\nExample NWSWSENE $$\\ =[3\\ 2\\ 1]=1+\\frac{1}{2+\\frac{1}{3}}=\\frac{10}{7}\\to\\text{ Parity: 0 }$$ NW SW SE NE Closures $\\ $ Closure Equivalence and pivoting to knots Theorem (Schubert) Suppose that rational tangles with fractions $\\frac{p}{q}$ and $\\frac{p^{\\prime}}{q^{\\prime}}$ are given ( $p$ and $q$ are relatively prime and $0\u0026lt;p$. Similarly for $p^{\\prime}$ and $q^{\\prime}$). If $N\\left(\\frac{p}{q}\\right)$ and $N\\left(\\frac{p^{\\prime}}{q^{\\prime}}\\right)$ denote the corresponding rational knots obtained by taking numerator closures of these tangles, then $N\\left(\\frac{p}{q}\\right)$ and $N\\left(\\frac{p^{\\prime}}{q^{\\prime}}\\right)$ are topologically equivalent if and only if (1) $p=p^{\\prime}$ (2) either $q \\equiv q^{\\prime}(\\bmod p)$ or $q q^{\\prime} \\equiv 1(\\bmod p)$. Schubert, Horst. \u0026ldquo;Knoten mit zwei Brücken..\u0026rdquo; Mathematische Zeitschrift 65 (1956): 133-170. http://eudml.org/doc/169591.\n\u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; 177175858%17=7\u0026#10; Montesinos Existence of canonical diagrams for Montesinos tangles Theorem (Bonahon and Siebenmann) Every non-rational Montesinos tangle $T$ admits a canonical diagram satisfying the following construction: $$T \\cong L_1+\\cdots+L_m+\\frac{k}{1}$$ where each $L_i \\cong \\frac{p_i}{q_i}$ is a rational subtangle in canonical form with fraction satisfying $0\u003c\\frac{p_i}{q_i}\u003c1$, and $\\frac{k}{1}$ is a horizontal integer subtangle. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\n$+$ $=$ $$\\ =\\ $$ $$=[3\\ 2\\ 0] + [3\\ 2\\ 0]$$ Generation For Montesinos tangles of crossing number $N$ we start by again generating twist vectors, however we require that each entry $e$ of the twist vector satisfies $2\\leq e \u0026lt; N.$\nWe call these restricted set of twist vectors stencils.\nFinding stencils for $N=5$ $$\\begin{array}{|l|l|l|l|} \\hline [1\\ 1\\ 1\\ 1\\ 1]\\ \u0026\\ [2\\ 1\\ 1\\ 1]\\ \u0026\\ [1\\ 2\\ 1\\ 1]\\ \u0026\\ [1\\ 1\\ 2\\ 1]\\\\\\hline [1\\ 1\\ 1\\ 2]\\ \u0026\\ [3\\ 1\\ 1]\\ \u0026\\ [1\\ 3\\ 1]\\ \u0026\\ [1\\ 1\\ 3]\\\\\\hline [2\\ 2\\ 1]\\ \u0026\\ [2\\ 1\\ 2]\\ \u0026\\ [1\\ 2\\ 2]\\ \u0026\\ [3\\ 2]\\\\\\hline [2\\ 3]\\ \u0026\\ [4\\ 1]\\ \u0026\\ [1\\ 4]\\ \u0026\\ [5]\\\\\\hline \\end{array}$$ Now for each entry $e_i$ of the stencil, we generate a list of rational tangles of crossing number equal to $e_i$, with the restriction $0\u0026lt;\\frac{p_i}{q_i}\u0026lt;1$. We then take all combinations of elements of these lists.\nMontesinos tangles for $N=5$ \\begin{array}{|l|} \\hline \\text{Rational Tangles CN: }2 \\\\\\hline [1\\ 1\\ 0]=\\frac{1}{2},\\ [2]=\\frac{2}{1} \\ \\\\\\hline \\text{Rational Tangles CN: }3\\\\\\hline [1\\ 2\\ 0]=\\frac{1}{3},\\ [2\\ 1\\ 0]=\\frac{2}{3},\\ [3]=\\frac{3}{1}\\\\\\hline \\end{array} $\\quad$ \\begin{array}{|l|l|} \\hline \\color{var(--r-Purple)}\\text{Stencil:}[3\\ 2]\\ \u0026\\ \\\\\\hline \\color{var(--r-Foreground)}[1\\ 2\\ 0] + [1\\ 1\\ 0]\\ \u0026\\ [2\\ 1\\ 0] + [1\\ 1\\ 0]\\\\\\hline \\color{var(--r-Purple)}\\text{Stencil:}[2\\ 3]\\\\\\hline \\color{var(--r-Foreground)}[1\\ 1\\ 0] + [1\\ 2\\ 0]\\ \u0026\\ [1\\ 1\\ 0] + [2\\ 1\\ 0]\\\\\\hline \\end{array} What about the \u0026lsquo;k\u0026rsquo;? The construction for the canonical Montesinos tangles includes a trailing $\\frac{k}{1}$ tangle. Our generation strategy seems to miss these.\nWhat we\u0026rsquo;re actually generating with this algorithm is Montesinos tangles up to moveable boundary components of the tangle. To recover fixed boundary tangles we can append an integral $k$ summand to each lower crossing Montesinos tangle with a circle product (more to come).\nProgrammatic Description stateDiagram-v2 direction LR state \u0026#34;For each stencil\u0026#34; as sten_loop{ state \u0026#34;Process stencil\u0026#34; as proc [*]--\u0026gt;proc proc --\u0026gt; [*] } [*]--\u0026gt; sten_loop sten_loop --\u0026gt; [*] stateDiagram-v2 state \u0026#34;while overflow false\u0026#34; as sten_loop{ direction LR state \u0026#34;For each array entry\u0026#34; as entry_loop{ direction LR state \u0026#34;Set overflow as false\u0026#34; as set_flw state \u0026#34;Increment entry\u0026#34; as inc_entry state \u0026#34;Set overflow as true\u0026#34; as over_true state \u0026#34;Set entry to zero\u0026#34; as zero_entry state \u0026#34;Break\u0026#34; as brk state overflow \u0026lt;\u0026lt;choice\u0026gt;\u0026gt; [*] --\u0026gt; set_flw set_flw --\u0026gt; inc_entry inc_entry --\u0026gt; overflow overflow --\u0026gt; zero_entry : if entry \u0026gt;= stencil entry overflow --\u0026gt; brk : if entry \u0026lt; stencil zero_entry --\u0026gt; over_true brk --\u0026gt; [*] over_true --\u0026gt; [*] } state \u0026#34;Get rational tangle\\nfor each array entry\u0026#34; as proc [*]--\u0026gt;proc proc --\u0026gt; entry_loop entry_loop --\u0026gt; [*] } state \u0026#34;Create len(stencil) array of all 0\u0026#34; as mk_ary state \u0026#34;Create overflow flag as false\u0026#34; as mk_flg [*]--\u0026gt; mk_ary mk_ary --\u0026gt; mk_flg mk_flg --\u0026gt; sten_loop sten_loop --\u0026gt; [*] stateDiagram-v2 direction LR state \u0026#34;For each stencil\u0026#34; as sten_loop{ state \u0026#34;Process stencil 1\u0026#34; as proc1 state \u0026#34;Process stencil 2\u0026#34; as proc2 state \u0026#34;Process stencil 3\u0026#34; as proc3 state \u0026#34;...\u0026#34; as proc4 state \u0026#34;Process stencil n\u0026#34; as proc5 state join_state \u0026lt;\u0026lt;join\u0026gt;\u0026gt; [*]--\u0026gt;proc1 [*]--\u0026gt;proc2 [*]--\u0026gt;proc3 [*]--\u0026gt;proc4 [*]--\u0026gt;proc5 proc1--\u0026gt;join_state proc2--\u0026gt;join_state proc3--\u0026gt;join_state proc4--\u0026gt;join_state proc5--\u0026gt;join_state join_state --\u0026gt; [*] } [*]--\u0026gt; sten_loop sten_loop --\u0026gt; [*] Using The Tanglenomicon Alpha To play with a live version, visit: \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e https://tanglenomicon.com\nWhere we\u0026rsquo;re going Generalized Montesinos Operation $\\circ$ $\\ $ $= \\color{var(--r-Purple)}([1\\ 2\\ 0] + [1\\ 2\\ 0] + [1\\ 1\\ 0]) \\color{var(--r-Foreground)}\\circ \\color{var(--r-Red)}[2\\ 2]$ Moon, Hyeyoung, and Isabel K. Darcy. \u0026ldquo;Tangle Equations Involving Montesinos Links.\u0026rdquo; Journal of Knot Theory and Its Ramifications 30, no. 08 (July 2021): 2150060. https://doi.org/10.1142/S0218216521500607.\nGeneration We just need to take our lists of Montesinos and rational tangles and glue them together with $\\circ$.\n$\\ $ $= \\color{var(--r-Purple)}([1\\ 2\\ 0] + [1\\ 2\\ 0] + [1\\ 1\\ 0]) \\color{var(--r-Foreground)}\\circ \\color{var(--r-Red)}[2\\ 2]$ Algebraic All possible tangles made from $+$ and $\\vee$ on basic tangles\nAlgebraic A tangle build from $\\vee$ and $+$ on some rational tangles. $$\\LP\\color{var(--r-Purple)}\\LB3\\ 2\\ 3\\RB+\\LB3\\ 2\\ 3\\RB\\color{var(--r-Foreground)}\\RP\\vee\\LP\\color{var(--r-Purple)}\\LB3\\ 2\\ 3\\RB+\\LB3\\ 2\\ 3\\RB\\color{var(--r-Foreground)}\\RP$$ Generation A tale of two strate-trees strate-tree 1 Algebraic Tangle Trees As we saw, we can linearize any algebraic tangle as:\n$$\\LP\\LB3\\ 2\\ 3\\RB+\\LB3\\ 2\\ 3\\RB\\RP\\vee\\LP\\LB3\\ 2\\ 3\\RB+\\LB3\\ 2\\ 3\\RB\\RP$$\nHow do we programmatically generate tangles from this?\n$$\\LP\\LB3\\ 2\\ 3\\RB+\\LB3\\ 2\\ 3\\RB\\RP\\vee\\LP\\LB3\\ 2\\ 3\\RB+\\LB3\\ 2\\ 3\\RB\\RP$$ [3 2 3][3 2 3][3 2 3][3 2 3]++v We can generate all possible algebraic expressions involving the basic tangles and twist vector of rational tangles.\nEquivalently, all full binary trees with $N$ leaves. Where the tree\u0026rsquo;s internal nodes are labeled with combinations of $\\vee$ and $+$ and leaves are labeled with all combinations of basic tangles or the twist vector of rational tangles.\nWe call these binary trees Algebraic Tangle Trees.\nAlain Caudron. Classification des nœuds et des enlacements, volume 4 of Publications Math ́ematiques d\u0026rsquo;Orsay 82 [Mathematical Publications of Orsay 82]. Universit ́e de ParisSud, D ́epartement de Mathe ́matique, Orsay, 1982. Connolly, Nicholas. Classification and Tabulation of 2-String Tangles: The Astronomy of Subtangle Decompositions. University of Iowa, 2021, https://doi.org/10.17077/etd.005978. A problem strate-tree 2 Arborescent Tangles Bonahon and Siebenmann describe a classification for what they call Arborescent Tangles. Their Arborescent Tangles can be translated into our algebraic tangles.\nThese Arborescent Tangles are constructed by taking a collection of twisted bands described by a weighted tree and connecting them with successive Murasugi sums.\nF. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\nx Y x Y 3 2 -3 0 4 3 Arborescent Tangles Generation ??? Into the future Non-algebraic/Polygonal 4-valent planar graphs $\\quad$ 4-valent planar graph insertions $6^*\\ *.[1\\ 2\\ 2\\ 3\\ 1].[1\\ 2\\ 2\\ 3\\ 1].[1\\ 2\\ 2\\ 3\\ 1].[1\\ 2\\ 2\\ 3\\ 1].[1\\ 2\\ 2\\ 3\\ 1]$ Generation There exist tables of 4 valent graphs. We can use those with insertions from our list of algebraic tangles to generate all polygonal tangles.\nTooling Design Goals The design for The Tanglenomicon project prioritizes flexibility and extensibility. We want a feature, maybe \u0026ldquo;calculate Jones polynomial,\u0026rdquo; to be runnable in a jupyter notebook or on a university cluster. We\u0026rsquo;re aiming for a \u0026ldquo;write once deploy anywhere\u0026rdquo; design.\nTo that end we\u0026rsquo;ve decoupled functionality wherever feasible, taking a layered approach for system design.\nflowchart LR Runner subgraph \u0026#34;Runnables\u0026#34; Generator Translator Computation end subgraph \u0026#34;Data Wranglers\u0026#34; Notation Storage end Runner --\u0026gt;|Runs| Generator Runner --\u0026gt;|Runs| Computation Runner --\u0026gt;|Runs| Translator Translator --\u0026gt;|Uses| Notation Generator --\u0026gt;|Uses| Notation Computation --\u0026gt;|Uses| Notation Generator --\u0026gt;|Uses| Storage Computation --\u0026gt;|Uses| Storage Translator --\u0026gt;|Uses| Storage Runners A runner is a human/machine interface layer. This abstracts the routines in lower layers for a user to interact with. This could be a CLI, Python binding, a Mathematica wrapper, or a web API.\nRunnables Generators\nGenerators create new data. A generator might look like a module to create rational tangles. They may use one or more Computations, Notations, or Translators.\nComputation\nComputations compute a value for a given data. A computation might look like a module for computing a Jones polynomial of a link, or computing the writhe of a tangle.\nTranslators\nTranslators define a conversion between two Notations. A translator might look like a module for converting from PD notation to Conway notation and back again.\nData Wranglers Notations\nNotations define a notational convention for a link/tangle. They describe a method for converting to and from a string representation of a link/tangle and data structure describing that link/tangle.\nStorage\nA storage module defines a storage interface for the application. The main inter-module type is string and the calling module is responsible for en/decoding the string with a notation module.\ncore libraries ------------------------------------------------------------------------------- Language files blank comment code ------------------------------------------------------------------------------- C 13 516 834 3563 C/C++ Header 18 408 967 1939 C++ 7 107 222 912 Markdown 21 418 0 794 SVG 5 5 5 322 CMake 41 74 21 236 TeX 1 1 0 92 Cython 1 21 2 83 JSON 2 1 0 78 Python 3 37 79 68 YAML 3 17 9 64 Nix 1 17 56 38 Text 1 0 0 7 ------------------------------------------------------------------------------- SUM: 117 1622 2195 8196 ------------------------------------------------------------------------------- Web API ┏━━━━━━━━━━━━━━━┳━━━━━━━┳━━━━━━━┳━━━━━━┳━━━━━━┳━━━━━━━━━┳━━━━━━┓ ┃ Language ┃ Files ┃ % ┃ Code ┃ % ┃ Comment ┃ % ┃ ┡━━━━━━━━━━━━━━━╇━━━━━━━╇━━━━━━━╇━━━━━━╇━━━━━━╇━━━━━━━━━╇━━━━━━┩ │ Python │ 27 │ 30.7 │ 1818 │ 53.9 │ 766 │ 22.7 │ │ Markdown │ 56 │ 63.6 │ 1473 │ 34.1 │ 0 │ 0.0 │ │ YAML │ 4 │ 4.5 │ 89 │ 93.7 │ 6 │ 6.3 │ │ __duplicate__ │ 1 │ 1.1 │ 0 │ 0.0 │ 0 │ 0.0 │ ├───────────────┼───────┼───────┼──────┼──────┼─────────┼──────┤ │ Sum │ 88 │ 100.0 │ 3380 │ 43.4 │ 772 │ 9.9 │ └───────────────┴───────┴───────┴──────┴──────┴─────────┴──────┘ Web frontend | language | files | code | comment | blank | total | |----------------------|-------|-------|---------|-------|-------| | JSON | 2 | 3,119 | 0 | 2 | 3,121 | | SVG | 1 | 1,489 | 1 | 2 | 1,492 | | JavaScript JSX | 6 | 398 | 9 | 43 | 450 | | source.markdown.math | 2 | 161 | 0 | 62 | 223 | | TypeScript JSX | 2 | 143 | 1 | 13 | 157 | | JavaScript | 7 | 141 | 2 | 14 | 157 | | XML | 5 | 56 | 0 | 0 | 56 | | JSON with Comments | 1 | 37 | 0 | 1 | 38 | | TypeScript | 1 | 20 | 0 | 4 | 24 | | CSS | 3 | 18 | 0 | 4 | 22 | | Nix | 1 | 16 | 0 | 3 | 19 | | HTML | 1 | 13 | 0 | 1 | 14 | | Docker | 1 | 12 | 1 | 11 | 24 | | Properties | 1 | 9 | 1 | 2 | 12 | Technologies ThrowTheSwitch/Unity Simple Unit Testing for C C 3.3k 935 Sources Dror Bar-Natan The Most Important Missing Infrastructure Project in Knot Theory Kauffman, L. H., and S. Lambropoulou. \u0026ldquo;From Tangle Fractions to DNA.\u0026rdquo; In Topology in Molecular Biology, edited by Michail Ilych Monastyrsky, 69-110. Biological and Medical Physics, Biomedical Engineering. Berlin, Heidelberg: Springer Berlin Heidelberg, 2007. https://doi.org/10.1007/978-3-540-49858-2_5. Moon, Hyeyoung, and Isabel K. Darcy. \u0026ldquo;Tangle Equations Involving Montesinos Links.\u0026rdquo; Journal of Knot Theory and Its Ramifications 30, no. 08 (July 2021): 2150060. https://doi.org/10.1142/S0218216521500607. Conway, J.H. \u0026ldquo;An Enumeration of Knots and Links, and Some of Their Algebraic Properties.\u0026rdquo; In Computational Problems in Abstract Algebra, 329-58. Elsevier, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5. Louis H. Kauffman and Sofia Lambropoulou. Classifying and applying rational knots and rational tangles. In DeTurck, editor, Contemporary Mathematics, volume 304, pages 223-259, 2001 Alain Caudron. Classification des nœuds et des enlacements, volume 4 of Publications Math ́ematiques d\u0026rsquo;Orsay 82 [Mathematical Publications of Orsay 82]. Universit ́e de ParisSud, D ́epartement de Mathe ́matique, Orsay, 1982. Robert Glenn Scharein. Interactive topological drawing. ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D. The University of British Columbia (Canada). URL: https://www.knotplot.com/. Jablan, S., \u0026amp; Sazdanović, R. (2007). Linknot. In Series on Knots and Everything. WORLD SCIENTIFIC. https://doi.org/10.1142/6623 Dowker, C. H., \u0026amp; Thistlethwaite, M. B. (1983). Classification of knot projections. In Topology and its Applications (Vol. 16, Issue 1, pp. 19-31). Elsevier BV. https://doi.org/10.1016/0166-8641(83)90004-4 Hoste, J., Thistlethwaite, M., \u0026amp; Weeks, J. (1998). The first 1,701,936 knots. In The Mathematical Intelligencer (Vol. 20, Issue 4, pp. 33-48). Springer Science and Business Media LLC. https://doi.org/10.1007/bf03025227 Burton, B. A. (2020). The Next 350 Million Knots. Schloss Dagstuhl - Leibniz-Zentrum Für Informatik. https://doi.org/10.4230/LIPICS.SOCG.2020.25 C. Livingston and A. H. Moore, KnotInfo: Table of Knot Invariants, knotinfo.math.indiana.edu, today\u0026rsquo;s date (eg. August 24, 2023). Schubert, Horst. \u0026ldquo;Knoten mit zwei Brücken..\u0026rdquo; Mathematische Zeitschrift 65 (1956): 133-170. http://eudml.org/doc/169591. Jos ́e M. Montesinos. Seifert manifolds that are ramified two-sheeted cyclic coverings. Bol. Soc. Mat. Mexicana (2), 18:1-32, 1973. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html Connolly, Nicholas. Classification and Tabulation of 2-String Tangles: The Astronomy of Subtangle Decompositions. University of Iowa, 2021, https://doi.org/10.17077/etd.005978. Sources Facebook, Public domain, via Wikimedia Commons FastAPI The MIT License (MIT) Carlos Baraza, CC0, via Wikimedia Commons Qq1040058283, Public domain, via Wikimedia Commons Jeremy Kratz, Public domain, via Wikimedia Commons Cython and Python, Apache License 2.0, via Wikimedia Commons mermaidjs www.python.org, GPL, via Wikimedia Commons Mongodb Ryan Dahl, MIT, via Wikimedia Commons Holger Krekel, CC BY 2.5, via Wikimedia Commons Alon Zakai, MIT, via Wikimedia Commons Cmake team. The original uploader was Francesco Betti Sorbelli at Italian Wikipedia.. Vectorized by Magasjukur2, CC BY 2.0, via Wikimedia Commons /* A left shift multiplies the value of an integer by 2. */ size_t count_lim = 0x01u \u0026lt;\u0026lt; (crossingNumber - 1); for (size_t i = 0u; i \u0026lt; count_lim; i++) { gen_rational_proc_template(i); } void proc_tmp(size_t template) { uint8_t counter = crossingNumber; size_t tv_length = 0; uint8_t twist_vector[UTIL_TANG_DEFS_MAX_CROSSINGNUM]={1}; while (counter \u0026gt; 0u) { counter--; if ((template \u0026amp; 0x01u) == 0) { tv_length++; } else { twist_vector[tv_length]++; } template = template \u0026gt;\u0026gt; 0x01u; } if (tv_length % 2 == 0) { evenperm_shift_write(); } else { write(); } } ","date":"19 August 2024","externalUrl":null,"permalink":"/speaking/research/comp_talk/","section":"Slides","summary":"Talk for my comprehensive exam on tangle tabulation.","title":"Comprehensive Exam Talk ","type":"slides"},{"content":" $\\newcommand{\\N}{\\mathbb{N}} \\newcommand{\\Z}{\\mathbb{Z}} \\newcommand{\\Q}{\\mathbb{Q}} \\newcommand{\\R}{\\mathbb{R}} \\newcommand{\\LP}{\\left(} \\newcommand{\\RP}{\\right)} \\newcommand{\\LS}{\\left\\lbrace} \\newcommand{\\RS}{\\right\\rbrace} \\newcommand{\\LA}{\\left\\langle} \\newcommand{\\RA}{\\right\\rangle} \\newcommand{\\LB}{\\left[} \\newcommand{\\RB}{\\right]} \\newcommand{\\MM}{\\ \\middle|\\ } \\newcommand{\\exp}{\\text{exp}} \\newcommand{\\abs}[1]{\\left\\vert#1\\right\\vert} \\newcommand{\\msr}[1]{m\\left(#1\\right)} \\newcommand{\\inv}[1]{#1^{-1}} \\newcommand{\\bkt}[1]{\\LA \\img{#1}\\RA} \\require{color}$ \u0026ldquo;What is?\u0026rdquo; is a selection of articles which explain an aspect of Math and/or Computer Science. The idea for \u0026ldquo;What is?\u0026rdquo; is taken from the immensely helpful list by Professor Nicholas J. Higham which was itself inspired by a similar series from the AMS.\nMy collection will deviate slightly from Prof. Higham\u0026rsquo;s list. I will try to explain each concept at three levels (where reasonable and I have time):\nGeneral - An acturate but not precise analogy understandable by a general reader. Some Training - An acturate but not precise analogy but with precision higer then that of the general level. Mature - An acturate and precise (or nearly so) description of the concept. Note: I\u0026rsquo;m using this series to build out a process for using generative AI.\nThis is the worst generative AI will ever be and I think it\u0026rsquo;s important we learn how it works and how to help our students use it effectively:\nWhen a section is AI generated I will include the emoji 🤖 in the section header.\nWhen a section is AI generated with manual edits I\u0026rsquo;ll include 🤖/👨.\nWhen a section is written wholey by myself I\u0026rsquo;ll include 👨.\n","date":"13 March 2024","externalUrl":null,"permalink":"/what_is/","section":"What is?","summary":"","title":"What is?","type":"What is?"},{"content":"","date":"13 March 2024","externalUrl":null,"permalink":"/series/what-is/","section":"Series","summary":"","title":"What Is?","type":"series"},{"content":" $\\newcommand{\\N}{\\mathbb{N}} \\newcommand{\\Z}{\\mathbb{Z}} \\newcommand{\\Q}{\\mathbb{Q}} \\newcommand{\\R}{\\mathbb{R}} \\newcommand{\\LP}{\\left(} \\newcommand{\\RP}{\\right)} \\newcommand{\\LS}{\\left\\lbrace} \\newcommand{\\RS}{\\right\\rbrace} \\newcommand{\\LA}{\\left\\langle} \\newcommand{\\RA}{\\right\\rangle} \\newcommand{\\LB}{\\left[} \\newcommand{\\RB}{\\right]} \\newcommand{\\MM}{\\ \\middle|\\ } \\newcommand{\\exp}{\\text{exp}} \\newcommand{\\abs}[1]{\\left\\vert#1\\right\\vert} \\newcommand{\\msr}[1]{m\\left(#1\\right)} \\newcommand{\\inv}[1]{#1^{-1}} \\newcommand{\\bkt}[1]{\\LA \\img{#1}\\RA} \\require{color}$ GEOTOP-A International Conference (1/11/24) The Tanglenomicon Zachary Bryhtan, Nicholas Connolly, Isabel Darcy, Ethan Rooke, Joseph Starr* Mathematics Department at The University of Iowa Knot Tables 1860\u0026rsquo;s Tait computes knots up to 7 crossing 15 knots 1870\u0026rsquo;s Tait, Kirkman, and Little compute knots up to 10 crossing Takes about 25 years 250 knots 1960\u0026rsquo;s Conway computes knots up to 11 crossings \u0026ldquo;A few hours\u0026rdquo; 802 knots 1980\u0026rsquo;s Dowker and Thistlethwaite compute up to 13 crossings First using a computer 12,966 1990\u0026rsquo;s Hoste, Thistlethwaite, and Weeks compute up to 16 crossings Computer runtime on the order of weeks 1,701,936 2020\u0026rsquo;s Burton computes up to 19 crossings 350 Million KnotInfo Conway How did Conway compute 25 years of work in \"a few hours\"? Tangles \u0026ldquo;We define a tangle as a portion of a knot diagram from which there emerge just 4 arcs pointing in the compass directions NW, NE, SW, SE.\u0026rdquo;\nConway, J.H. \u0026ldquo;An Enumeration of Knots and Links, and Some of Their Algebraic Properties.\u0026rdquo; In Computational Problems in Abstract Algebra, 329-58. Elsevier, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5\n$\\quad$ $\\quad$ NWNESWSE $\\quad$ $\\quad$ $\\quad$ Basic Operations Operation $+$ $+$ $=$ $=$ $=$ $2$ Operation $\\vee$ $\\vee$ $=$ $=$ $=$ $\\frac{1}{2}$ The Tanglenomicon The table of two string tangles Building up $\\ $ $\\ $ $\\ $ $\\ $ Where we are Rational Tangles $\\ $ $\\begin{aligned}\\to\u0026\\ \\LP 3 \\vee \\frac{1}{2}\\RP + 2\\\\\u0026\\\\ \\to\u0026\\ [3\\ 2\\ 2]\\end{aligned}$ Generation For any $N$ an obvious twist vector is the twist vector of all $1$s $$[1\\ 1\\ 1\\ \\cdots\\ 1]$$ Noting that when we write this sequence, we have $N-1$ spaces.\nIf we choose to place a $+$ instead of the left most space we get $$[1+1\\ 1\\ \\cdots\\ 1]=[2\\ 1\\ \\cdots\\ 1]$$ we\u0026rsquo;re free to make this choice for each space\nthis gives $N-1$ choices between \u0026lsquo;$+$\u0026rsquo; and space $$[1\\square 1\\square 1\\square\\cdots\\square1]$$ letting us generate twist vectors by simply counting from $0\\to 2^{N-1}$.\nTwist Vectors for $N=5$ $$\\begin{array}{|l|l|l|l|} \\hline [1\\ 1\\ 1\\ 1\\ 1]\\ \u0026\\ [2\\ 1\\ 1\\ 1]\\ \u0026\\ [1\\ 2\\ 1\\ 1]\\ \u0026\\ [1\\ 1\\ 2\\ 1]\\\\\\hline [1\\ 1\\ 1\\ 2]\\ \u0026\\ [3\\ 1\\ 1]\\ \u0026\\ [1\\ 3\\ 1]\\ \u0026\\ [1\\ 1\\ 3]\\\\\\hline [2\\ 2\\ 1]\\ \u0026\\ [2\\ 1\\ 2]\\ \u0026\\ [1\\ 2\\ 2]\\ \u0026\\ [3\\ 2]\\\\\\hline [2\\ 3]\\ \u0026\\ [4\\ 1]\\ \u0026\\ [1\\ 4]\\ \u0026\\ [5]\\\\\\hline \\end{array}$$ Canonical Twist Vectors We can write a canonical twist vector by taking the odd length vectors (appending $0$ where needed).\nCanonical Twist Vectors for $N=5$ $$\\begin{array}{|l|l|l|l|} \\hline [1\\ 1\\ 1\\ 1\\ 1]\\ \u0026\\ [2\\ 1\\ 1\\ 1\\ 0]\\ \u0026\\ [1\\ 2\\ 1\\ 1\\ 0]\\ \u0026\\ [1\\ 1\\ 2\\ 1\\ 0]\\\\\\hline [1\\ 1\\ 1\\ 2\\ 0]\\ \u0026\\ [3\\ 1\\ 1]\\ \u0026\\ [1\\ 3\\ 1]\\ \u0026\\ [1\\ 1\\ 3]\\\\\\hline [2\\ 2\\ 1]\\ \u0026\\ [2\\ 1\\ 2]\\ \u0026\\ [1\\ 2\\ 2]\\ \u0026\\ [3\\ 2\\ 0]\\\\\\hline [2\\ 3\\ 0]\\ \u0026\\ [4\\ 1\\ 0]\\ \u0026\\ [1\\ 4\\ 0]\\ \u0026\\ [5]\\\\\\hline \\end{array}$$ Computations Rational Number (continued fraction) The rational number for a twist vector is computed by taking the twist vector as a finite continued fraction that is: $$\\LB a\\ b\\ c\\RB=c+\\frac{1}{b+\\frac{1}{a}}$$\nTwist Vector to rational number $$\\ =\\LB 3\\ 2\\ 2\\RB=2+\\frac{1}{2+\\frac{1}{3}}=\\frac{17}{7}$$ Louis H. Kauffman and Sofia Lambropoulou. Classifying and applying rational knots and rational tangles. In DeTurck, editor, Contemporary Mathematics, volume 304, pages 223-259, 2001\nUsing The Tanglenomicon Where we\u0026rsquo;re going Montesinos Existence of canonical diagrams for Montesinos tangles Theorem (Bonahon and Siebenmann) Every non-rational Montesinos tangle $T$ admits a canonical diagram satisfying the following construction: $$T \\cong L_1+\\cdots+L_m+\\frac{k}{1}$$ where each $L_i \\cong \\frac{p_i}{q_i}$ is a rational subtangle in canonical form with fraction satisfying $0\u003c\\frac{p_i}{q_i}\u003c1$, and $\\frac{k}{1}$ is a horizontal integer subtangle. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\n$+$ $=$ $$\\ =\\ $$ $$=[3\\ 2\\ 0] + [3\\ 2\\ 0]$$ Generation The Montesinos tangles of crossing number $N$ have a slightly simpler generation strategy compared to rational tangles. We again generate twist vectors but require that each entry $e$ of the twist vector satisfies $2\\leq e \u0026lt; N.$ We call these restricted set of twist vectors stencils.\nStencils for $N=5$ $$\\begin{array}{|l|l|l|l|} \\hline [1\\ 1\\ 1\\ 1\\ 1]\\ \u0026\\ [2\\ 1\\ 1\\ 1]\\ \u0026\\ [1\\ 2\\ 1\\ 1]\\ \u0026\\ [1\\ 1\\ 2\\ 1]\\\\\\hline [1\\ 1\\ 1\\ 2]\\ \u0026\\ [3\\ 1\\ 1]\\ \u0026\\ [1\\ 3\\ 1]\\ \u0026\\ [1\\ 1\\ 3]\\\\\\hline [2\\ 2\\ 1]\\ \u0026\\ [2\\ 1\\ 2]\\ \u0026\\ [1\\ 2\\ 2]\\ \u0026\\ [3\\ 2]\\\\\\hline [2\\ 3]\\ \u0026\\ [4\\ 1]\\ \u0026\\ [1\\ 4]\\ \u0026\\ [5]\\\\\\hline \\end{array}$$ Now for each entry $e_i$ of the stencil, we generate a list of rational tangles of crossing number equal to $e_i$, with the restriction $0\u0026lt;\\frac{p_i}{q_i}\u0026lt;1$. We then take all combinations of elements of these lists.\nMontesinos tangles for $N=5$ \\begin{array}{|l|} \\hline \\text{Rational Tangles CN: }2 \\\\\\hline [1\\ 1\\ 0]=\\frac{1}{2},\\ [2]=\\frac{2}{1} \\ \\\\\\hline \\text{Rational Tangles CN: }3\\\\\\hline [1\\ 2\\ 0]=\\frac{1}{3},\\ [2\\ 1\\ 0]=\\frac{2}{3},\\ [3]=\\frac{3}{1}\\\\\\hline \\end{array} $\\quad$ \\begin{array}{|l|l|} \\hline \\color{var(--r-Purple)}\\text{Stencil:}[3\\ 2]\\ \u0026\\ \\\\\\hline \\color{var(--r-Foreground)}[1\\ 2\\ 0] + [1\\ 1\\ 0]\\ \u0026\\ [2\\ 1\\ 0] + [1\\ 1\\ 0]\\\\\\hline \\color{var(--r-Purple)}\\text{Stencil:}[2\\ 3]\\\\\\hline \\color{var(--r-Foreground)}[1\\ 1\\ 0] + [1\\ 2\\ 0]\\ \u0026\\ [1\\ 1\\ 0] + [2\\ 1\\ 0]\\\\\\hline \\end{array} What about the \u0026lsquo;k\u0026rsquo;? The construction for the canonical Montesinos tangles includes a trailing $\\frac{k}{1}$ tangle. Our generation strategy seems to miss these.\nWhat we\u0026rsquo;re actually generating with this algorithm is equivalent to allowing the boundary components of the tangle to move. To recover fixed boundary tangles we need to generate the next larger class of tangles.\nWe can note which stage a tangle was generated at to allow users to choose datasets for fixed or non-fixed boundary tangles.\nGeneralized Montesinos Operation $\\circ$ $\\ $ $= \\color{var(--r-Purple)}([1\\ 2\\ 0] + [1\\ 2\\ 0] + [1\\ 1\\ 0]) \\color{var(--r-Foreground)}\\circ \\color{var(--r-Red)}[1\\ 2]$ Moon, Hyeyoung, and Isabel K. Darcy. \u0026ldquo;Tangle Equations Involving Montesinos Links.\u0026rdquo; Journal of Knot Theory and Its Ramifications 30, no. 08 (July 2021): 2150060. https://doi.org/10.1142/S0218216521500607.\nGeneration We just need to take our lists of Montesinos and rational tangles and glue them together with $\\circ$.\nInto the future Algebraic All possible tangles made from $+$ and $\\vee$\nAlgebraic A vertical sum of two Montesinos tangles. Generation Algebraic Tangle Trees To generate all possible algebraic tangles, we can generate all possible algebraic expressions on the trivial tangles. Equivalently, all full binary trees with $N$ leaves. Where the tree\u0026rsquo;s internal nodes are labeled with combinations of $\\vee$ and $+$ and leaves are labeled with all combinations of trivial tangles.\nThese binary trees are called Algebraic Tangle Trees.\nAlain Caudron. Classification des nœuds et des enlacements, volume 4 of Publications Math ́ematiques d\u0026rsquo;Orsay 82 [Mathematical Publications of Orsay 82]. Universit ́e de ParisSud, D ́epartement de Mathe ́matique, Orsay, 1982.\nConnolly, Nicholas. Classification and Tabulation of 2-String Tangles: The Astronomy of Subtangle Decompositions. University of Iowa, 2021, https://doi.org/10.17077/etd.005978.\n[3 2 3][3 2 3][3 2 3][3 2 3]++v Non-algebraic/Polygonal 4-valent planar graphs $\\quad$ 4-valent planar graph insertions $6^*\\ *.[1\\ 2\\ 2\\ 3\\ 1].[1\\ 2\\ 2\\ 3\\ 1].[1\\ 2\\ 2\\ 3\\ 1].[1\\ 2\\ 2\\ 3\\ 1].[1\\ 2\\ 2\\ 3\\ 1]$ Generation There exist tables of 4 valent graphs. We can use those with insertions from our list of algebraic tangles to generate all polygonal tangles.\nTechnologies ThrowTheSwitch/Unity Simple Unit Testing for C C 3.3k 935 Sources Dror Bar-Natan The Most Important Missing Infrastructure Project in Knot Theory Kauffman, L. H., and S. Lambropoulou. \u0026ldquo;From Tangle Fractions to DNA.\u0026rdquo; In Topology in Molecular Biology, edited by Michail Ilych Monastyrsky, 69-110. Biological and Medical Physics, Biomedical Engineering. Berlin, Heidelberg: Springer Berlin Heidelberg, 2007. https://doi.org/10.1007/978-3-540-49858-2_5. Moon, Hyeyoung, and Isabel K. Darcy. \u0026ldquo;Tangle Equations Involving Montesinos Links.\u0026rdquo; Journal of Knot Theory and Its Ramifications 30, no. 08 (July 2021): 2150060. https://doi.org/10.1142/S0218216521500607. Conway, J.H. \u0026ldquo;An Enumeration of Knots and Links, and Some of Their Algebraic Properties.\u0026rdquo; In Computational Problems in Abstract Algebra, 329-58. Elsevier, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5. Louis H. Kauffman and Sofia Lambropoulou. Classifying and applying rational knots and rational tangles. In DeTurck, editor, Contemporary Mathematics, volume 304, pages 223-259, 2001 Alain Caudron. Classification des nœuds et des enlacements, volume 4 of Publications Math ́ematiques d\u0026rsquo;Orsay 82 [Mathematical Publications of Orsay 82]. Universit ́e de ParisSud, D ́epartement de Mathe ́matique, Orsay, 1982. Robert Glenn Scharein. Interactive topological drawing. ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D. The University of British Columbia (Canada). URL: https://www.knotplot.com/. Jablan, S., \u0026amp; Sazdanović, R. (2007). Linknot. In Series on Knots and Everything. WORLD SCIENTIFIC. https://doi.org/10.1142/6623 Dowker, C. H., \u0026amp; Thistlethwaite, M. B. (1983). Classification of knot projections. In Topology and its Applications (Vol. 16, Issue 1, pp. 19-31). Elsevier BV. https://doi.org/10.1016/0166-8641(83)90004-4 Hoste, J., Thistlethwaite, M., \u0026amp; Weeks, J. (1998). The first 1,701,936 knots. In The Mathematical Intelligencer (Vol. 20, Issue 4, pp. 33-48). Springer Science and Business Media LLC. https://doi.org/10.1007/bf03025227 Burton, B. A. (2020). The Next 350 Million Knots. Schloss Dagstuhl - Leibniz-Zentrum Für Informatik. https://doi.org/10.4230/LIPICS.SOCG.2020.25 C. Livingston and A. H. Moore, KnotInfo: Table of Knot Invariants, knotinfo.math.indiana.edu, today\u0026rsquo;s date (eg. August 24, 2023). Schubert, Horst. \u0026ldquo;Knoten mit zwei Brücken..\u0026rdquo; Mathematische Zeitschrift 65 (1956): 133-170. http://eudml.org/doc/169591. Jos ́e M. Montesinos. Seifert manifolds that are ramified two-sheeted cyclic coverings. Bol. Soc. Mat. Mexicana (2), 18:1-32, 1973. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html Connolly, Nicholas. Classification and Tabulation of 2-String Tangles: The Astronomy of Subtangle Decompositions. University of Iowa, 2021, https://doi.org/10.17077/etd.005978. Programmatic Description stateDiagram-v2 direction LR state if_done \u0026lt;\u0026lt;choice\u0026gt;\u0026gt; State_i: i=0 State_ipp: i\u0026#43;\u0026#43; state \u0026#34;Construct TV from i as a bitfield\u0026#34; as tv_calc{ state \u0026#34;tmp=i;j=0;cnt=N\u0026#34; as State_temp State_jpp: j\u0026#43;\u0026#43; State_cntmm: cnt-- State_sum_tv: TV[j]\u0026#43;\u0026#43; State_rsh: tmp=tmp\u0026gt;\u0026gt;1 state if_lsb \u0026lt;\u0026lt;choice\u0026gt;\u0026gt; state if_cnteo \u0026lt;\u0026lt;choice\u0026gt;\u0026gt; State_store_tv: Store TV [*] --\u0026gt; State_temp State_temp --\u0026gt; if_cnteo if_cnteo--\u0026gt; State_cntmm: if cnt\u0026gt;0 if_cnteo--\u0026gt; State_store_tv: if cnt==0 State_store_tv --\u0026gt; [*] State_cntmm --\u0026gt;if_lsb if_lsb --\u0026gt;State_sum_tv: if (tmp \u0026amp; 0x01u)==1u State_sum_tv --\u0026gt; State_rsh if_lsb --\u0026gt;State_jpp: if (tmp \u0026amp; 0x01u)==0u State_jpp --\u0026gt; State_rsh State_rsh --\u0026gt; if_cnteo } [*] --\u0026gt; State_i State_i --\u0026gt; if_done if_done --\u0026gt; tv_calc: if i \u0026lt; 2**(N-1) tv_calc --\u0026gt; State_ipp State_ipp --\u0026gt; if_done if_done --\u0026gt; [*]: if i == 2**(N-1) Computations Rational Number (continued fraction) The rational number for a twist vector is computed by taking the twist vector as a finite continued fraction that is: $$\\LB a\\ b\\ c\\RB=c+\\frac{1}{b+\\frac{1}{a}}$$\nTwist Vector to rational number $$\\ =\\LB 3\\ 2\\ 2\\RB=2+\\frac{1}{2+\\frac{1}{3}}=\\frac{17}{7}$$ Louis H. Kauffman and Sofia Lambropoulou. Classifying and applying rational knots and rational tangles. In DeTurck, editor, Contemporary Mathematics, volume 304, pages 223-259, 2001\nTo play with twist vectors and continued fractions visit\n\u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e https://joe-starr.com/resources/cont_frac_convert/\nParity NWSWSENE NW SW SE NE NWSWSENE NW SW SE NE NWSWSENE NW SW SE NE Computing Parity If we take the rational number $\\frac{p}{q}$ associated with the rational tangle we get the following correspondence for parity\nParity Table $$\\begin{array}{|c|c|c|} \\hline p\\ \\%\\ 2 \u0026q\\ \\%\\ 2\u0026\\text{Parity}\\\\ \\hline 0 \u00260\u0026N/A\\\\ \\hline 0 \u00261\u0026 0 \\\\ \\hline 1 \u00260\u0026\\infty\\\\ \\hline 1 \u00261\u0026 1\\\\ \\hline \\end{array}$$ Note NWSWSENE $$\\ =[3\\ 2\\ 1]=1+\\frac{1}{2+\\frac{1}{3}}=\\frac{10}{7}\\to\\text{ Parity: 0 }$$ NW SW SE NE Closures $\\ $ Closure Equivalence and pivoting to knots Theorem (Schubert) Suppose that rational tangles with fractions $\\frac{p}{q}$ and $\\frac{p^{\\prime}}{q^{\\prime}}$ are given ( $p$ and $q$ are relatively prime and $0$\u003c$p$. Similarly for $p^{\\prime}$ and $q^{\\prime}$.) If $K\\left(\\frac{p}{q}\\right)$ and $K\\left(\\frac{p^{\\prime}}{q^{\\prime}}\\right)$ denote the corresponding rational knots obtained by taking numerator closures of these tangles, then $K\\left(\\frac{p}{q}\\right)$ and $K\\left(\\frac{p^{\\prime}}{q^{\\prime}}\\right)$ are topologically equivalent if and only if (1) $p=p^{\\prime}$ (2) either $q \\equiv q^{\\prime}(\\bmod p)$ or $q q^{\\prime} \\equiv 1(\\bmod p)$. Schubert, Horst. \u0026ldquo;Knoten mit zwei Brücken..\u0026rdquo; Mathematische Zeitschrift 65 (1956): 133-170. http://eudml.org/doc/169591.\n\u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; 177175858%17=7\u0026#10; Tooling Design Goals The design for The Tanglenomicon project prioritizes flexibility and extensibility. We want a feature, maybe \u0026ldquo;calculate Jones polynomial,\u0026rdquo; to be runnable in a jupyter notebook or on a university cluster. We\u0026rsquo;re aiming for a \u0026ldquo;write once deploy anywhere\u0026rdquo; design.\nTo that end we\u0026rsquo;ve decoupled functionality wherever feasible, taking a layered approach for system design.\nflowchart LR Runner subgraph \u0026#34;Runnables\u0026#34; Generator Translator Computation end subgraph \u0026#34;Data Wranglers\u0026#34; Notation Storage end Runner --\u0026gt;|Runs| Generator Runner --\u0026gt;|Runs| Computation Runner --\u0026gt;|Runs| Translator Translator --\u0026gt;|Uses| Notation Generator --\u0026gt;|Uses| Notation Computation --\u0026gt;|Uses| Notation Generator --\u0026gt;|Uses| Storage Computation --\u0026gt;|Uses| Storage Translator --\u0026gt;|Uses| Storage Runners A runner is a human/machine interface layer. This abstracts the routines in lower layers for a user to interact with. This could be a CLI, python binding, a Mathematica wrapper, or a web API.\nRunnables Generators\nGenerators create new data. A generator might look like a module to create rational tangles. They may use one or more Computations, Notations, or Translators.\nComputation\nComputations compute a value for a given data. A computation might look like a module for computing a Jones polynomial of a link, or a computing the writhe of a tangle.\nTranslators\nTranslators define a conversion between two Notations. A translator might look like a module for converting from PD notation to Conway notation and back again.\nData Wranglers Notations\nNotations define a notational convention for a link/tangle. They describe a method for converting to and from a string representation of a link/tangle and data structure describing that link/tangle.\nStorage\nA storage module defines a storage interface for the application. The main inter-module type is string and the calling module is responsible for en/decoding the string with a notation module.\nParallelization sequenceDiagram participant DB participant Server participant Client 1 participant Client 2 Server-\u0026gt;\u0026gt;\u0026#43;Client 1: Dispatch job 1 for stencil starting from TV idx\u0026lt;br/\u0026gt;[0,0,0,...] Server--\u0026gt;\u0026gt;DB: Mark job 1 as dispatched Server-\u0026gt;\u0026gt;\u0026#43;Client 2: Dispatch job 2 for stencil starting from TV idx\u0026lt;br/\u0026gt;[100,0,0,...] Server--\u0026gt;\u0026gt;DB: Mark job 2 as dispatched Client 1--\u0026gt;\u0026gt;-Server: job complete Server--\u0026gt;\u0026gt;DB: store job 1 results and mark complete Server-\u0026gt;\u0026gt;\u0026#43;Client 1: Dispatch job 3 for stencil starting from TV idx\u0026lt;br/\u0026gt;[0,100,0,...] Server--\u0026gt;\u0026gt;DB: Mark job 3 as dispatched Client 2--\u0026gt;\u0026gt;-Server: job complete Server--\u0026gt;\u0026gt;DB: store job 2 results and mark complete Client 1--\u0026gt;\u0026gt;-Server: job complete Server--\u0026gt;\u0026gt;DB: store job 3 results and mark complete ","date":"11 January 2024","externalUrl":null,"permalink":"/speaking/research/merida_24/","section":"Slides","summary":"Talk given at the GEOTOP-A International Conference on 1/11/24.","title":"GEOTOP-A International Conference Applications of Geometry and Topology","type":"slides"},{"content":" $\\newcommand{\\N}{\\mathbb{N}} \\newcommand{\\Z}{\\mathbb{Z}} \\newcommand{\\Q}{\\mathbb{Q}} \\newcommand{\\R}{\\mathbb{R}} \\newcommand{\\LP}{\\left(} \\newcommand{\\RP}{\\right)} \\newcommand{\\LS}{\\left\\lbrace} \\newcommand{\\RS}{\\right\\rbrace} \\newcommand{\\LA}{\\left\\langle} \\newcommand{\\RA}{\\right\\rangle} \\newcommand{\\LB}{\\left[} \\newcommand{\\RB}{\\right]} \\newcommand{\\MM}{\\ \\middle|\\ } \\newcommand{\\exp}{\\text{exp}} \\newcommand{\\abs}[1]{\\left\\vert#1\\right\\vert} \\newcommand{\\msr}[1]{m\\left(#1\\right)} \\newcommand{\\inv}[1]{#1^{-1}} \\newcommand{\\bkt}[1]{\\LA \\img{#1}\\RA} \\require{color}$ Zombies and knots Constructing the Jones polynomial to save the world $\\ $\nDay 0: smalltowns-ville, IA: A man picks up his daily slice of breakfast pizza from his local gas station. What he doesn\u0026rsquo;t know is that it\u0026rsquo;s his last. By mid-day he feels terrible, by the time he\u0026rsquo;s ready to go home for dinner he\u0026rsquo;s already feasting on brains.\nDay 4: CDC Headquarters The Z-virus has spread midwest-wide. You\u0026rsquo;re working at the CDC as an expert in microscopy. You\u0026rsquo;re working frantically to get any information on the Z-virus you can. In furtherance of that goal, you decide to image the DNA of the Z-Virus.\nDNA knot as seen under the electron microscope. - Image Credit: Javier Arsuaga, CC BY-ND\nDNA Deoxyribonucleic acid (abbreviated DNA) is the molecule that carries genetic information for the development and functioning of an organism.\nDNA is made of two linked strands that wind around each other to resemble a twisted ladder — a shape known as a double helix.\nDeoxyribonucleic acid (DNA). (n.d.). Genome.gov. https://www.genome.gov/genetics-glossary/Deoxyribonucleic-Acid. Accessed 3 October 2023\n\u0026#1092;\u0026#1086;\u0026#1089;\u0026#1092;\u0026#1072;\u0026#1090;\u0026#1076;\u0026#1077;\u0026#1079;\u0026#1086;\u0026#1082;\u0026#1089;\u0026#1080;\u0026#1088;\u0026#1080;\u0026#1073;\u0026#1086;\u0026#1079;\u0026#1072;\u0026#1086;\u0026#1089;\u0026#1085;\u0026#1086;\u0026#1074;\u0026#1085;\u0026#1072;\u0026#1103; \u0026#1094;\u0026#1077;\u0026#1087;\u0026#1100; Phosphat-Deoxyriboser\u0026#252;ckgrat Khung x\u0026#432;\u0026#417;ngPhosphat-deoxyribose Phosphate-deoxyribosebackbone \u0026#1040;\u0026#1076;\u0026#1077;\u0026#1085;\u0026#1080;\u0026#1085; Adenin A\u0026#273;\u0026#234;nin Adenine \u0026#1062;\u0026#1080;\u0026#1090;\u0026#1086;\u0026#1079;\u0026#1080;\u0026#1085; Cytosin Xit\u0026#244;zin Cytosine \u0026#1043;\u0026#1091;\u0026#1072;\u0026#1085;\u0026#1080;\u0026#1085; Guanin Guanin Guanine \u0026#1058;\u0026#1080;\u0026#1084;\u0026#1080;\u0026#1085; Thymin Timin Thymine O O P O \u0026#8722;O O \u0026#8722; OH O O O\u0026#8722; P O O OH \u0026#8722; NH2 N N N N H2N N N N N NH2 N N O H2N N N O O NH N O O HN N O O N NH N N NH2 O N HN N N H2N 3'-\u0026#1082;\u0026#1086;\u0026#1085;\u0026#1077;\u0026#1094; 3\u0026#8242; Ende \u0026#272;\u0026#7847;u 3' 3\u0026#8242; end 5'-\u0026#1082;\u0026#1086;\u0026#1085;\u0026#1077;\u0026#1094; 5\u0026#8242; Ende \u0026#272;\u0026#7847;u 5' 5\u0026#8242; end 3'-\u0026#1082;\u0026#1086;\u0026#1085;\u0026#1077;\u0026#1094; 3\u0026#8242; Ende \u0026#272;\u0026#7847;u 3' 3\u0026#8242; end 5'-\u0026#1082;\u0026#1086;\u0026#1085;\u0026#1077;\u0026#1094; 5\u0026#8242; Ende \u0026#272;\u0026#7847;u 5' 5\u0026#8242; end Each strand has a backbone. Attached to each sugar is one of four bases: adenine (A), cytosine (C), guanine (G) or thymine (T). The two strands are connected by chemical bonds between the bases: adenine bonds with thymine, and cytosine bonds with guanine. Deoxyribonucleic acid (DNA). (n.d.). Genome.gov. https://www.genome.gov/genetics-glossary/Deoxyribonucleic-Acid. Accessed 3 October 2023\nPhoto by Madprime (talk · contribs) - This vector image was created with Inkscape ., CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=1848174\nMacro Structure Circular DNA Dulbecco and Vogt (1963) and Weil and Vinograd (1963) discovered that double-stranded DNA of the polyoma virus is circular.\nVologodskii, A. V. (n.d.). Circular DNA. In Cyclic Polymers (pp. 47-83). Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47117-5_2\n\u0026ldquo;Supercoiled\u0026rdquo; DNA Vinograd, J., Lebowitz, J., Radloff, R., Watson, R., \u0026amp; Laipis, P. (1965) discover that double-stranded DNA can \u0026ldquo;supercoil\u0026rdquo;.\nVinograd, J., Lebowitz, J., Radloff, R., Watson, R., \u0026amp; Laipis, P. (1965). The twisted circular form of polyoma viral DNA. In Proceedings of the National Academy of Sciences (Vol. 53, Issue 5, pp. 1104-1111). Proceedings of the National Academy of Sciences. https://doi.org/10.1073/pnas.53.5.1104\nDay 7: CDC Headquarters The spread is now nation wide but still under some control.\nYou\u0026rsquo;ve successfully imaged the DNA of the Z-virus and found DNA with a knot. Your CDC coworkers are using your findings to construct an anti-Z-virus. The anti-virus is the mirror of the DNA knot you\u0026rsquo;ve found. This will allow the human body to build anti-bodies for the Z-virus.\nThe CDC now needs you to verify that the DNA knot they\u0026rsquo;ve produced truly is the mirror of the Z-virus.\nAnti-Knot $\\ $ Mathematical Knots \u0026ldquo;A knot is a smooth embedding of a circle $S^1$ into Euclidean 3-dimensional space $\\R^3$ (or the 3-dimensional sphere $S^3$ ).\u0026rdquo;\n$\\quad$ $\\quad$ $\\quad$ Jablan, S., \u0026amp; Sazdanović, R. (2007). Linknot. In Series on Knots and Everything. WORLD SCIENTIFIC. https://doi.org/10.1142/6623\nhttps://www.knotplot.com/\nDiagrams for knotted dna $\\to$ $\\to$ DNA knot as seen under the electron microscope. - Image Credit: Javier Arsuaga, CC BY-ND\nKnot Equivalence Reidemeister moves Type I $\\leftrightarrow$ Type II \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e $\\leftrightarrow$ \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e Type III \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e $\\leftrightarrow$ \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e Equal? $\\ $ Playing with diagrams What\u0026rsquo;s the important information inside a knot diagram?\n$\\,$ \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e $\\ $ Clockwise \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e $\\to$ anti-clockwise \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e $\\to$ \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e Pictures are hard lets leverage algebra $\\bkt{/presentations/kauf_bkt/crossing/crossing_un.svg}$ Skein Relation ${\\begin{matrix} \\ \u0026\\left.CW\\right.\u0026 \u0026 \\left.CCW\\right.\u0026\\\\ \\ \u0026\\left.\\img{/presentations/kauf_bkt/crossing/crossing_un.svg}\\right.\u0026 \u0026\\left.\\img{/presentations/kauf_bkt/crossing/crossing_un.svg}\\right.\u0026\\\\ \\ \u0026\\left.\\downarrow \\right.\u0026 \u0026\\left. \\downarrow \\right.\u0026\\\\ \\bkt{/presentations/kauf_bkt/crossing/crossing_un.svg}=\\ A\u0026\\LA \\img{/presentations/kauf_bkt/type2/6a.svg} \\RA\u0026+B\u0026\\LA \\img{/presentations/kauf_bkt/type2/6b.svg}\\RA\u0026 \\end{matrix}}$ What are we looking for? We want to use our bracket to build a polynomial that can tell two knots apart. In particular, we want to differentiate a knot and its \u0026ldquo;anti-knot\u0026rdquo;(mirror).\nPutting pieces together How can we tell two knots apart? How can we use that and our bracket to build our polynomial? Check what happens under Reidemeister moves If our bracket \u0026ldquo;respects\u0026rdquo; Reidemeister moves it respects knot \u0026ldquo;equivalence\u0026rdquo;.\nType II ${\\LA\\img{/presentations/kauf_bkt/type2/1.svg}\\RA=\\LA\\img{/presentations/kauf_bkt/type2/6b.svg}\\RA}$ $ \\small{\\LA \\img{/presentations/kauf_bkt/crossing/crossing_un.svg}\\RA=A\\LA \\img{/presentations/kauf_bkt/type2/6a.svg} \\RA+B\\LA\\img{/presentations/kauf_bkt/type2/6b.svg}\\RA}$ $\\begin{aligned} \\bkt{/presentations/kauf_bkt/type2/1.svg} \u0026=A\\bkt{/presentations/kauf_bkt/type2/2a.svg}+B\\bkt{/presentations/kauf_bkt/type2/2b.svg}\\\\ \u0026=A \\LP A\\bkt{/presentations/kauf_bkt/type2/3a.svg}+B\\bkt{/presentations/kauf_bkt/type2/4.svg}\\RP\\\\ \u0026+B\\LP A\\bkt{/presentations/kauf_bkt/type2/6b.svg}+B\\bkt{/presentations/kauf_bkt/type2/6a.svg}\\RP\\\\ \\end{aligned}$ A problem ${B\\bkt{/presentations/kauf_bkt/type2/4.svg}}$ $\\begin{aligned} 1.\\quad\u0026{\\LA \\img{/presentations/kauf_bkt/unknot.svg} \\RA=1}\\\\ 2.\\quad\u0026{\\LA P\\sqcup \\img{/presentations/kauf_bkt/unknot.svg} \\RA=C\\LA P\\RA} \\end{aligned}$ Back to computing $\\begin{aligned} A\\LP A\\bkt{/presentations/kauf_bkt/type2/3a.svg}+B\\bkt{/presentations/kauf_bkt/type2/4.svg}\\RP \u0026+B\\LP A\\bkt{/presentations/kauf_bkt/type2/6b.svg}+B\\bkt{/presentations/kauf_bkt/type2/6a.svg}\\RP\\\\ \u0026=A\\LP A\\bkt{/presentations/kauf_bkt/type2/6a.svg}+BC\\bkt{/presentations/kauf_bkt/type2/6a.svg}\\RP\\\\ \u0026+B\\LP A\\bkt{/presentations/kauf_bkt/type2/6b.svg}+B\\bkt{/presentations/kauf_bkt/type2/6a.svg}\\RP\\\\ \\end{aligned}$ $\\begin{aligned} \u0026=A^2\\bkt{/presentations/kauf_bkt/type2/6a.svg}+ABC\\bkt{/presentations/kauf_bkt/type2/6a.svg}\\\\ \u0026+BA\\bkt{/presentations/kauf_bkt/type2/6b.svg}+B^2\\bkt{/presentations/kauf_bkt/type2/6a.svg}\\\\ \u0026=\\LP A^2+ABC+B^2\\RP\\bkt{/presentations/kauf_bkt/type2/6a.svg}\\\\ \u0026+BA\\bkt{/presentations/kauf_bkt/type2/6b.svg} \\end{aligned}$ What we wanted ${\\LA\\img{/presentations/kauf_bkt/type2/1.svg}\\RA=\\LA\\img{/presentations/kauf_bkt/type2/6b.svg}\\RA}$ What we have $\\LA\\img{/presentations/kauf_bkt/type2/1.svg}\\RA=\\LP A^2+ABC+B^2\\RP\\bkt{/presentations/kauf_bkt/type2/6a.svg}+BA\\bkt{/presentations/kauf_bkt/type2/6b.svg}$ So we need $\\LP A^2+ABC+B^2\\RP\\bkt{/presentations/kauf_bkt/type2/6a.svg}+BA\\bkt{/presentations/kauf_bkt/type2/6b.svg}=\\LA\\img{/presentations/kauf_bkt/type2/6b.svg}\\RA$ Putting pieces together How can we select $A,\\ B$, and $C$ to get equality?\n$\\LP A^2+ABC+B^2\\RP\\bkt{/presentations/kauf_bkt/type2/6a.svg}+BA\\bkt{/presentations/kauf_bkt/type2/6b.svg}=\\LA\\img{/presentations/kauf_bkt/type2/6b.svg}\\RA$ $B=\\inv{A}$ $\\begin{aligned} \\LP A^2+ABC+B^2\\RP\\bkt{/presentations/kauf_bkt/type2/6a.svg}+BA\\bkt{/presentations/kauf_bkt/type2/6b.svg}\u0026=\\LA\\img{/presentations/kauf_bkt/type2/6b.svg}\\RA\\\\ \\LP A^2+C+A^{-2}\\RP\\bkt{/presentations/kauf_bkt/type2/6a.svg}+\\bkt{/presentations/kauf_bkt/type2/6b.svg}\u0026=\\LA\\img{/presentations/kauf_bkt/type2/6b.svg}\\RA\\\\ \\end{aligned}$ $C=-A^{-2}-A^2$ $\\begin{aligned} \\LP A^2+C+A^{-2}\\RP\\bkt{/presentations/kauf_bkt/type2/6a.svg}+\\bkt{/presentations/kauf_bkt/type2/6b.svg}\u0026=\\LA\\img{/presentations/kauf_bkt/type2/6b.svg}\\RA\\\\ \\bkt{/presentations/kauf_bkt/type2/6b.svg}\u0026=\\LA\\img{/presentations/kauf_bkt/type2/6b.svg}\\RA\\\\ \\end{aligned}$ Type II ${\\LA\\img{/presentations/kauf_bkt/type2/1.svg}\\RA=\\LA\\img{/presentations/kauf_bkt/type2/6b.svg}\\RA}$ $ \\begin{aligned} 1.\\quad\u0026\\LA \\img{/presentations/kauf_bkt/crossing/crossing_un.svg}\\RA=A\\LA \\img{/presentations/kauf_bkt/type2/6a.svg} \\RA+\\inv{A}\\LA\\img{/presentations/kauf_bkt/type2/6b.svg}\\RA\\\\ 2.\\quad\u0026{\\LA \\img{/presentations/kauf_bkt/unknot.svg} \\RA=1}\\\\ 3.\\quad\u0026{\\LA P\\sqcup \\img{/presentations/kauf_bkt/unknot.svg} \\RA=\\LP-A^{-2}-A^2\\RP\\LA P\\RA} \\end{aligned}$ Exercise: Type III $\\LA\\img{/presentations/kauf_bkt/type3/1.svg}\\RA=\\LA\\img{/presentations/kauf_bkt/type3/6.svg}\\RA$ Type I $\\LA\\img{/presentations/kauf_bkt/type1/1.svg}\\RA$ $\\LA\\img{/presentations/kauf_bkt/type1/1b.svg}\\RA$ $\\begin{aligned} \\LA\\img{/presentations/kauf_bkt/type1/1b.svg}\\RA\u0026= A\\LA\\img{/presentations/kauf_bkt/type1/2a.svg}\\RA+A^{-1}\\LA\\img{/presentations/kauf_bkt/type1/2b.svg}\\RA\\\\ \u0026=A\\LA\\img{/presentations/kauf_bkt/type1/2a.svg}\\RA\\\\ \u0026+A^{-1}\\LP -A^{-2}-A^2\\RP\\LA\\img{/presentations/kauf_bkt/type1/2a.svg}\\RA\\\\ \\end{aligned}$ $\\begin{aligned} \\LA\\img{/presentations/kauf_bkt/type1/1b.svg}\\RA\u0026=A\\LA\\img{/presentations/kauf_bkt/type1/2a.svg}\\RA\\\\ \u0026+A^{-1}\\LP -A^{-2}-A^2\\RP\\LA\\img{/presentations/kauf_bkt/type1/2a.svg}\\RA\\\\ \u0026=\\LP A-A^{-3}-A\\RP\\LA\\img{/presentations/kauf_bkt/type1/2a.svg}\\RA\\\\ \\end{aligned}$ $${\\LA\\img{/presentations/kauf_bkt/type1/1b.svg}\\RA=-A^{-3}\\LA\\img{/presentations/kauf_bkt/type1/2a.svg}\\RA}$$ Exercise: Compute bracket for the other Type I ${\\LA\\img{/presentations/kauf_bkt/type1/1.svg}\\RA=?\\LA\\img{/presentations/kauf_bkt/type1/2a.svg}\\RA}$ ${\\LA\\img{/presentations/kauf_bkt/type1/1.svg}\\RA=-A^{3}\\LA\\img{/presentations/kauf_bkt/type1/2a.svg}\\RA}$ Why is this a problem? ${\\begin{aligned} \\LA\\img{/presentations/kauf_bkt/type1/1b.svg}\\RA\u0026=-A^{-3}\\LA\\img{/presentations/kauf_bkt/type1/2a.svg}\\RA\\\\ \\LA\\img{/presentations/kauf_bkt/type1/1.svg}\\RA\u0026=-A^{3}\\LA\\img{/presentations/kauf_bkt/type1/2a.svg}\\RA \\end{aligned} } $ What do we have so far? For Type II and III everything \u0026ldquo;works\u0026rdquo; with the rules:\n$ \\begin{aligned} 1.\\quad\u0026{\\LA \\img{/presentations/kauf_bkt/unknot.svg} \\RA=1}\\\\ 2.\\quad\u0026\\LA \\img{/presentations/kauf_bkt/crossing/crossing_un.svg}\\RA=A\\LA \\img{/presentations/kauf_bkt/type2/6a.svg} \\RA+\\inv{A}\\LA\\img{/presentations/kauf_bkt/type2/6b.svg}\\RA\\\\ 3.\\quad\u0026{\\LA P\\sqcup \\img{/presentations/kauf_bkt/unknot.svg} \\RA=\\LP-A^{-2}-A^2\\RP\\LA P\\RA} \\end{aligned}$ but Type I is \u0026ldquo;broken\u0026rdquo;:\n${\\LA\\img{/presentations/kauf_bkt/type1/1b.svg}\\RA=-A^{-3}\\LA\\img{/presentations/kauf_bkt/type1/2a.svg}\\RA\\quad\\LA\\img{/presentations/kauf_bkt/type1/1.svg}\\RA=-A^{3}\\LA\\img{/presentations/kauf_bkt/type1/2a.svg}\\RA}$ Day 53 Time is running out. With your preliminary results in hand the vaccine is being produced. The future of the world is now on your shoulders waiting for your results.\nHow can we fix Type I $ \\begin{aligned} \\LA\\img{/presentations/kauf_bkt/type1/1b.svg}\\RA\u0026=-A^{-3}\\LA\\img{/presentations/kauf_bkt/type1/2a.svg}\\RA\\\\ \\LA\\img{/presentations/kauf_bkt/type1/1.svg}\\RA\u0026=-A^{3}\\LA\\img{/presentations/kauf_bkt/type1/2a.svg}\\RA \\end{aligned}$ Orientation of a crossing $1.$ Positive $\\quad$ $2.$ Negative Writhe of a knot The writhe $w\\LP P\\RP$ of a diagram $P$ of an oriented link is the sum of the signs of the crossings of $ P $.\n$${w\\LP P\\RP=\\text{#}\\LP\\img{/presentations/kauf_bkt/crossing/crossing_+.svg}\\RP-\\text{#}\\LP\\img{/presentations/kauf_bkt/crossing/crossing_-.svg}\\RP}$$ Exercise: Compute the writhe $w\\LP\\img{/presentations/kauf_bkt/trefoil/trefoil.svg}\\RP$ Fixing Type I $ -A^{-3w\\LP \\img{/presentations/kauf_bkt/type1/1b.svg}\\RP}\\LA\\img{/presentations/kauf_bkt/type1/1b.svg}\\RA $ $ \\begin{aligned} -A^{-3w\\LP \\img{/presentations/kauf_bkt/type1/1b.svg}\\RP}\\LA\\img{/presentations/kauf_bkt/type1/1b.svg}\\RA \u0026= -A^{-3\\LP-1\\RP}\\LP-A^{-3}\\RP\\LA\\img{/presentations/kauf_bkt/type1/2a.svg}\\RA\\\\ \u0026= -A^{3}\\LP-A^{-3}\\RP\\LA\\img{/presentations/kauf_bkt/type1/2a.svg}\\RA\\\\ \u0026=\\LA\\img{/presentations/kauf_bkt/type1/2a.svg}\\RA \\end{aligned} $ Exercise: Verify the other type I move $-A^{-3w\\LP \\img{/presentations/kauf_bkt/type1/1.svg}\\RP}=\\LA\\img{/presentations/kauf_bkt/type1/2a.svg}\\RA$ What do we have? For Type I, II, and III everything \u0026ldquo;works\u0026rdquo; for the polynomial $$V\\LP P\\RP=-A^{-3w\\LP P\\RP}\\LA P\\RA$$ with the rules:\n$ \\begin{aligned} 1.\\quad\u0026{\\LA \\img{/presentations/kauf_bkt/unknot.svg} \\RA=1}\\\\ 2.\\quad\u0026\\LA \\img{/presentations/kauf_bkt/crossing/crossing_un.svg}\\RA=A\\LA \\img{/presentations/kauf_bkt/type2/6a.svg} \\RA+\\inv{A}\\LA\\img{/presentations/kauf_bkt/type2/6b.svg}\\RA\\\\ 3.\\quad\u0026{\\LA P\\sqcup \\img{/presentations/kauf_bkt/unknot.svg} \\RA=\\LP-A^{-2}-A^2\\RP\\LA P\\RA} \\end{aligned}$ We can now compute $V\\LP\\img{/presentations/kauf_bkt/trefoil/trefoil.svg}\\RP=-A^{-3w\\LP \\img{/presentations/kauf_bkt/trefoil/trefoil.svg}\\RP}\\bkt{/presentations/kauf_bkt/trefoil/trefoil.svg}$ $-A^{-3w\\LP \\img{/presentations/kauf_bkt/trefoil/trefoil.svg}\\RP}\\bkt{/presentations/kauf_bkt/trefoil/trefoil.svg}$ $ \\begin{aligned} \u0026=-A^{-3\\cdot -3}\\LP A\\bkt{/presentations/kauf_bkt/trefoil/trefoil_a.svg} +\\inv{A}\\bkt{/presentations/kauf_bkt/trefoil/trefoil_b.svg}\\RP\\\\ \u0026=-A^{9}\\LP A\\bkt{/presentations/kauf_bkt/trefoil/trefoil_a.svg}+\\inv{A}\\bkt{/presentations/kauf_bkt/trefoil/trefoil_b.svg}\\RP\\\\ \\end{aligned} $ $ \\begin{aligned} \\bkt{/presentations/kauf_bkt/trefoil/trefoil_a.svg} \u0026=-A^{3}\\bkt{/presentations/kauf_bkt/trefoil/trefoil_ab.svg}\\\\ \u0026=-A^{3}\\LP-A^{3}\\RP\\bkt{/presentations/kauf_bkt/unknot.svg}\\\\ \u0026=A^{6}\\\\ \\end{aligned} $ $-A^{-3w\\LP \\img{/presentations/kauf_bkt/trefoil/trefoil.svg}\\RP}\\bkt{/presentations/kauf_bkt/trefoil/trefoil.svg}$ $ \\begin{aligned} \u0026-A^{9}\\LP A\\bkt{/presentations/kauf_bkt/trefoil/trefoil_a.svg}+\\inv{A}\\bkt{/presentations/kauf_bkt/trefoil/trefoil_b.svg}\\RP\\\\ \u0026= -A^{9}\\LP A\\LP A^6\\RP+\\inv{A}\\bkt{/presentations/kauf_bkt/trefoil/trefoil_b.svg}\\RP\\\\ \\end{aligned} $ $ \\begin{aligned} \\bkt{/presentations/kauf_bkt/trefoil/trefoil_b.svg} \u0026=A\\bkt{/presentations/kauf_bkt/trefoil/trefoil_ba.svg} +A^{-1}\\bkt{/presentations/kauf_bkt/trefoil/trefoil_bb.svg}\\\\ \u0026=A\\LP -A^{3}\\bkt{/presentations/kauf_bkt/unknot.svg}\\RP\\\\ \u0026+A^{-1}\\LP -A^{-3}\\bkt{/presentations/kauf_bkt/unknot.svg}\\RP\\\\ \u0026=-A^{4}-A^{-4}\\\\ \\end{aligned} $ $ \\begin{aligned} \u0026-A^{-3w\\LP \\img{/presentations/kauf_bkt/trefoil/trefoil.svg}\\RP}\\bkt{/presentations/kauf_bkt/trefoil/trefoil.svg}\\\\ \u0026=-A^{9}\\LP A\\bkt{/presentations/kauf_bkt/trefoil/trefoil_a.svg}+\\inv{A}\\bkt{/presentations/kauf_bkt/trefoil/trefoil_b.svg}\\RP\\\\ \u0026= -A^{9}\\LP A\\LP A^6\\RP+\\inv{A}\\bkt{/presentations/kauf_bkt/trefoil/trefoil_b.svg}\\RP\\\\ \u0026= -A^{9}\\LP A\\LP A^6\\RP+\\inv{A}\\LP-A^{4}-A^{-4}\\RP\\RP\\\\ \u0026= -A^{9}\\LP A^7-A^{3}-A^{-5}\\RP\\\\ \u0026= -A^{16}+A^{12}+A^{4}\\\\ \\end{aligned} $ Exercise: Compute the bracket on the anti-knot $-A^{-3w\\LP \\img{/presentations/dna/dna_right.svg}\\RP}\\LA\\img{/presentations/dna/dna_right.svg}\\RA$ Anti-Knot ${ -A^{16}+A^{12}+A^{4}}$ $\\quad$ ${ -A^{-16}+A^{-12}+A^{-4}}$ Day 121 With the successful completion of your work the vaccine is being administer world wide. The President congratulates you for your work and the world is optimistic.\nDay 300 The virus is completely controlled and you win every prize in every field imaginable!\nThe Jones Polynomial The Jones Polynomial $V\\LP \\mathscr{K}\\RP$ of an oriented knot $\\mathscr{K}$ is the Laurent polynomial with integer coefficients in $t^{1/2}$.\nDefined by $ V\\LP \\mathscr{K}\\RP=\\LP\\LP-A\\RP^{-3w(P)}\\LA P\\RA\\RP _{t^{1/2}=A^{-2}} $ where $P$ is any oriented diagram for $\\mathscr{K}$.\n$ \\begin{aligned} V\\LP \\mathscr{K}\\RP\u0026= \\LP-A^{-3 w\\LP\\img{/presentations/kauf_bkt/trefoil/trefoil.svg}\\RP} \\bkt{/presentations/kauf_bkt/trefoil/trefoil.svg}\\RP _{t^{1/2}=A^{-2}}\\\\ \u0026=\\LP-A^{-3\\cdot-3} \\bkt{/presentations/kauf_bkt/trefoil/trefoil.svg}\\RP _{t^{1/2}=A^{-2}}\\\\ \u0026=\\LP-A^{9}\\LP A^7-A^3-A^{-5}\\RP\\RP _{t^{1/2}=A^{-2}}\\\\ \u0026=\\LP-A^{16}+A^{12}+A^{-4}\\RP _{t^{1/2}=A^{-2}}\\\\ \u0026=-t^{-4}+t^{-3}+t^{-1}\\\\ \\end{aligned} $ Worksheet\nLivingston, C. (1993). Knot Theory. Mathematical Association of America. https://doi.org/10.5948/UPO9781614440239 Dale Rolfsen, Knots and links, Mathematics Lecture Series, vol. 7, Publish or Perish, Inc., Houston, TX, 1990, Corrected reprint of the 1976 original. Robert Glenn Scharein. Interactive topological drawing. ProQuest LLC, Ann Arbor, MI, 1998. Thesis Ph.D. The University of British Columbia (Canada). URL: https://www.knotplot.com/. Jablan, S., \u0026amp; Sazdanović, R. (2007). Linknot. In Series on Knots and Everything. WORLD SCIENTIFIC. https://doi.org/10.1142/6623 Vaughan Jones. The Jones polynomial for dummies. https://math.berkeley.edu/~vfr/jonesakl.pdf WebArchive Deoxyribonucleic acid (DNA). (n.d.). Genome.gov. https://www.genome.gov/genetics-glossary/Deoxyribonucleic-Acid. Accessed 3 October 2023 DNA knot as seen under the electron microscope. - Image Credit: Javier Arsuaga, CC BY-ND Vinograd, J., Lebowitz, J., Radloff, R., Watson, R., \u0026amp; Laipis, P. (1965). The twisted circular form of polyoma viral DNA. In Proceedings of the National Academy of Sciences (Vol. 53, Issue 5, pp. 1104-1111). Proceedings of the National Academy of Sciences. https://doi.org/10.1073/pnas.53.5.1104 Photo by Madprime (talk · contribs) - This vector image was created with Inkscape ., CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=1848174 ","date":"17 November 2023","externalUrl":null,"permalink":"/speaking/general_audience/zombie_jones/","section":"Slides","summary":"In this talk we give a construction of the Jones Polynomial via the Kauffman bracket. The concepts are introduced in a pseudo-application of fighting a zombie apocalypse.","title":"Constructing the Jones polynomial to save the world","type":"slides"},{"content":"","date":"17 November 2023","externalUrl":null,"permalink":"/speaking/general_audience/","section":"Slides","summary":"","title":"General Audience","type":"speaking"},{"content":"","date":"17 November 2023","externalUrl":null,"permalink":"/tags/general-audience/","section":"Tags","summary":"","title":"General Audience","type":"tags"},{"content":" $\\newcommand{\\N}{\\mathbb{N}} \\newcommand{\\Z}{\\mathbb{Z}} \\newcommand{\\Q}{\\mathbb{Q}} \\newcommand{\\R}{\\mathbb{R}} \\newcommand{\\LP}{\\left(} \\newcommand{\\RP}{\\right)} \\newcommand{\\LS}{\\left\\lbrace} \\newcommand{\\RS}{\\right\\rbrace} \\newcommand{\\LA}{\\left\\langle} \\newcommand{\\RA}{\\right\\rangle} \\newcommand{\\LB}{\\left[} \\newcommand{\\RB}{\\right]} \\newcommand{\\MM}{\\ \\middle|\\ } \\newcommand{\\exp}{\\text{exp}} \\newcommand{\\abs}[1]{\\left\\vert#1\\right\\vert} \\newcommand{\\msr}[1]{m\\left(#1\\right)} \\newcommand{\\inv}[1]{#1^{-1}} \\newcommand{\\bkt}[1]{\\LA \\img{#1}\\RA} \\require{color}$ U of M - Dearborn Colloquium (11/15/23) The Tanglenomicon Zachary Bryhtan, Nicholas Connolly, Isabel Darcy, Ethan Rooke, Joseph Starr* Mathematics Department at The University of Iowa Knots \u0026ldquo;A knot is a smooth embedding of a circle $S^1$ into Euclidean 3-dimensional space $\\R^3$ (or the 3-dimensional sphere $S^3$ ).\u0026rdquo;\n$\\quad$ $\\quad$ $\\quad$ Jablan, S., \u0026amp; Sazdanović, R. (2007). Linknot. In Series on Knots and Everything. WORLD SCIENTIFIC. https://doi.org/10.1142/6623\nhttps://www.knotplot.com/\nKnot Equivalence reidemeister moves Type I $\\leftrightarrow$ Type II \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e $\\leftrightarrow$ \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e Type III \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e $\\leftrightarrow$ \u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e The natural question Knot Tables Lord Kelvin\u0026rsquo;s vortex theory of the atom. Atoms are knotted vortices in the æther. 1860\u0026rsquo;s Tait computes knots up to 7 crossing 15 knots 1870\u0026rsquo;s Tait, Kirkman, and Little compute knots up to 10 crossing Takes about 25 years 250 knots 1960\u0026rsquo;s Conway computes knots up to 11 crossings \u0026ldquo;A few hours\u0026rdquo; 802 knots 1980\u0026rsquo;s Dowker and Thistlethwaite compute up to 13 crossings First using a computer 12,966 1990\u0026rsquo;s Hoste, Thistlethwaite, and Weeks compute up to 16 crossings Computer runtime on the order of weeks 1,701,936 2020\u0026rsquo;s Burton computes up to 19 crossings 350 Million KnotInfo Conway How did Conway compute 25 years of work in \"a few hours\"? Tangles \u0026ldquo;We define a tangle as a portion of a knot diagram from which there emerge just 4 arcs pointing in the compass directions NW, NE, SW, SE.\u0026rdquo;\nConway, J.H. \u0026ldquo;An Enumeration of Knots and Links, and Some of Their Algebraic Properties.\u0026rdquo; In Computational Problems in Abstract Algebra, 329-58. Elsevier, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5\n$\\quad$ $\\quad$ NWNESWSE $\\quad$ $\\quad$ $\\quad$ Basic Operations Operation $+$ $+$ $=$ $=$ $=$ $2$ Operation $\\vee$ $\\vee$ $=$ $=$ $=$ $\\frac{1}{2}$ The Tanglenomicon Building up $\\ $ $\\ $ $\\ $ $\\ $ Where we are Rational Tangles $\\ $ $\\begin{aligned}\\to\u0026\\ \\LP 3 \\vee \\frac{1}{2}\\RP + 2\\\\\u0026\\\\ \\to\u0026\\ [3\\ 2\\ 2]\\end{aligned}$ Generation For any $N$ an obvious twist vector is the twist vector of all $1$s $$[1\\ 1\\ 1\\ \\cdots\\ 1]$$ Noting that when we write this sequence we have $N-1$ spaces.\nIf we choose to place a $+$ instead of the left most space we get $$[1+1\\ 1\\ \\cdots\\ 1]=[2\\ 1\\ \\cdots\\ 1]$$ we\u0026rsquo;re free to make this choice for each space\nthis gives $N-1$ choices between \u0026lsquo;$+$\u0026rsquo; and space $$[1\\square 1\\square 1\\square\\cdots\\square1]$$ letting us generate twist vectors by simply counting from $0\\to 2^{N-1}$.\nTwist Vectors for $N=5$ $$\\begin{array}{|l|l|l|l|} \\hline [1\\ 1\\ 1\\ 1\\ 1]\\ \u0026\\ [2\\ 1\\ 1\\ 1]\\ \u0026\\ [1\\ 2\\ 1\\ 1]\\ \u0026\\ [1\\ 1\\ 2\\ 1]\\\\\\hline [1\\ 1\\ 1\\ 2]\\ \u0026\\ [3\\ 1\\ 1]\\ \u0026\\ [1\\ 3\\ 1]\\ \u0026\\ [1\\ 1\\ 3]\\\\\\hline [2\\ 2\\ 1]\\ \u0026\\ [2\\ 1\\ 2]\\ \u0026\\ [1\\ 2\\ 2]\\ \u0026\\ [3\\ 2]\\\\\\hline [2\\ 3]\\ \u0026\\ [4\\ 1]\\ \u0026\\ [1\\ 4]\\ \u0026\\ [5]\\\\\\hline \\end{array}$$ Canonical Twist Vectors We can write a canonical twist vector by taking the odd length vectors (appending $0$ where needed).\nCanonical Twist Vectors for $N=5$ $$\\begin{array}{|l|l|l|l|} \\hline [1\\ 1\\ 1\\ 1\\ 1]\\ \u0026\\ [2\\ 1\\ 1\\ 1\\ 0]\\ \u0026\\ [1\\ 2\\ 1\\ 1\\ 0]\\ \u0026\\ [1\\ 1\\ 2\\ 1\\ 0]\\\\\\hline [1\\ 1\\ 1\\ 2\\ 0]\\ \u0026\\ [3\\ 1\\ 1]\\ \u0026\\ [1\\ 3\\ 1]\\ \u0026\\ [1\\ 1\\ 3]\\\\\\hline [2\\ 2\\ 1]\\ \u0026\\ [2\\ 1\\ 2]\\ \u0026\\ [1\\ 2\\ 2]\\ \u0026\\ [3\\ 2\\ 0]\\\\\\hline [2\\ 3\\ 0]\\ \u0026\\ [4\\ 1\\ 0]\\ \u0026\\ [1\\ 4\\ 0]\\ \u0026\\ [5]\\\\\\hline \\end{array}$$ Programmatic Description stateDiagram-v2 direction LR state if_done \u0026lt;\u0026lt;choice\u0026gt;\u0026gt; State_i: i=0 State_ipp: i\u0026#43;\u0026#43; state \u0026#34;Construct TV from i as a bitfield\u0026#34; as tv_calc{ state \u0026#34;tmp=i;j=0;cnt=N\u0026#34; as State_temp State_jpp: j\u0026#43;\u0026#43; State_cntmm: cnt-- State_sum_tv: TV[j]\u0026#43;\u0026#43; State_rsh: tmp=tmp\u0026gt;\u0026gt;1 state if_lsb \u0026lt;\u0026lt;choice\u0026gt;\u0026gt; state if_cnteo \u0026lt;\u0026lt;choice\u0026gt;\u0026gt; State_store_tv: Store TV [*] --\u0026gt; State_temp State_temp --\u0026gt; if_cnteo if_cnteo--\u0026gt; State_cntmm: if cnt\u0026gt;0 if_cnteo--\u0026gt; State_store_tv: if cnt==0 State_store_tv --\u0026gt; [*] State_cntmm --\u0026gt;if_lsb if_lsb --\u0026gt;State_sum_tv: if (tmp \u0026amp; 0x01u)==1u State_sum_tv --\u0026gt; State_rsh if_lsb --\u0026gt;State_jpp: if (tmp \u0026amp; 0x01u)==0u State_jpp --\u0026gt; State_rsh State_rsh --\u0026gt; if_cnteo } [*] --\u0026gt; State_i State_i --\u0026gt; if_done if_done --\u0026gt; tv_calc: if i \u0026lt; 2**(N-1) tv_calc --\u0026gt; State_ipp State_ipp --\u0026gt; if_done if_done --\u0026gt; [*]: if i == 2**(N-1) Computations Rational Number (continued fraction) The rational number for a twist vector is computed by taking the twist vector as a finite continued fraction that is: $$\\LB a\\ b\\ c\\RB=c+\\frac{1}{b+\\frac{1}{a}}$$\nTwist Vector to rational number $$\\ =\\LB 3\\ 2\\ 2\\RB=2+\\frac{1}{2+\\frac{1}{3}}=\\frac{17}{7}$$ Louis H. Kauffman and Sofia Lambropoulou. Classifying and applying rational knots and rational tangles. In DeTurck, editor, Contemporary Mathematics, volume 304, pages 223-259, 2001\nTo play with twist vectors and continued fractions visit\n\u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e https://joe-starr.com/resources/cont_frac_convert/\nParity NWSWSENE NW SW SE NE NWSWSENE NW SW SE NE Computing Parity If we take the rational number $\\frac{p}{q}$ associated with the rational tangle we get the following correspondence for parity\nParity Table $$\\begin{array}{|c|c|c|} \\hline p\\ \\%\\ 2 \u0026q\\ \\%\\ 2\u0026\\text{Parity}\\\\ \\hline 0 \u00260\u0026N/A\\\\ \\hline 0 \u00261\u0026 0 \\\\ \\hline 1 \u00260\u0026\\infty\\\\ \\hline 1 \u00261\u0026 1\\\\ \\hline \\end{array}$$ Note NWSWSENE $$\\ =[3\\ 2\\ 1]=1+\\frac{1}{2+\\frac{1}{3}}=\\frac{10}{7}\\to\\text{ Parity: 0 }$$ NW SW SE NE Closures $\\ $ Closure Equivalence and pivoting to knots Theorem (Schubert) Suppose that rational tangles with fractions $\\frac{p}{q}$ and $\\frac{p^{\\prime}}{q^{\\prime}}$ are given ( $p$ and $q$ are relatively prime and $0$\u003c$p$. Similarly for $p^{\\prime}$ and $q^{\\prime}$.) If $K\\left(\\frac{p}{q}\\right)$ and $K\\left(\\frac{p^{\\prime}}{q^{\\prime}}\\right)$ denote the corresponding rational knots obtained by taking numerator closures of these tangles, then $K\\left(\\frac{p}{q}\\right)$ and $K\\left(\\frac{p^{\\prime}}{q^{\\prime}}\\right)$ are topologically equivalent if and only if (1) $p=p^{\\prime}$ (2) either $q \\equiv q^{\\prime}(\\bmod p)$ or $q q^{\\prime} \\equiv 1(\\bmod p)$. Schubert, Horst. \u0026ldquo;Knoten mit zwei Brücken..\u0026rdquo; Mathematische Zeitschrift 65 (1956): 133-170. http://eudml.org/doc/169591.\n\u003c?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?\u003e \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; \u0026#10; 177175858%17=7\u0026#10; Using The Tanglenomicon Where we\u0026rsquo;re going Montesinos Existence of canonical diagrams for Montesinos tangles Theorem (Bonahon and Siebenmann) Every non-rational Montesinos tangle $T$ admits a canonical diagram satisfying the following construction: $$T \\cong L_1+\\cdots+L_m+\\frac{k}{1}$$ where each $L_i \\cong \\frac{p_i}{q_i}$ is a rational subtangle in canonical form with fraction satisfying $0\u003c\\frac{p_i}{q_i}\u003c1$, and $\\frac{k}{1}$ is a horizontal integer subtangle. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html\n$+$ $=$ $$\\ =\\ $$ $$=[3\\ 2\\ 2] + [3\\ 2\\ 2]$$ Generation The Montesinos tangles of crossing number $N$ have a slightly simpler generation strategy compared to rational tangles. We again generate twist vectors but require that each entry $e$ of the twist vector satisfies $2\\leq e \u0026lt; N.$ We call these restricted set of twist vectors stencils.\nStencils for $N=5$ $$\\begin{array}{|l|l|l|l|} \\hline [1\\ 1\\ 1\\ 1\\ 1]\\ \u0026\\ [2\\ 1\\ 1\\ 1]\\ \u0026\\ [1\\ 2\\ 1\\ 1]\\ \u0026\\ [1\\ 1\\ 2\\ 1]\\\\\\hline [1\\ 1\\ 1\\ 2]\\ \u0026\\ [3\\ 1\\ 1]\\ \u0026\\ [1\\ 3\\ 1]\\ \u0026\\ [1\\ 1\\ 3]\\\\\\hline [2\\ 2\\ 1]\\ \u0026\\ [2\\ 1\\ 2]\\ \u0026\\ [1\\ 2\\ 2]\\ \u0026\\ [3\\ 2]\\\\\\hline [2\\ 3]\\ \u0026\\ [4\\ 1]\\ \u0026\\ [1\\ 4]\\ \u0026\\ [5]\\\\\\hline \\end{array}$$ Now for each entry $e_i$ of the stencil, we generate a list of rational tangles of crossing number equal to $e_i$, with the restriction $0\u0026lt;\\frac{p_i}{q_i}\u0026lt;1$. We then take all combinations of elements of these lists.\nMontesinos tangles for $N=5$ \\begin{array}{|l|} \\hline \\text{Rational Tangles CN: }2 \\\\\\hline [1\\ 1\\ 0]=\\frac{1}{2},\\ [2]=\\frac{2}{1} \\ \\\\\\hline \\text{Rational Tangles CN: }3\\\\\\hline [1\\ 2\\ 0]=\\frac{1}{3},\\ [2\\ 1\\ 0]=\\frac{2}{3},\\ [3]=\\frac{3}{1}\\\\\\hline \\end{array} $\\quad$ \\begin{array}{|l|l|} \\hline \\color{var(--r-Purple)}\\text{Stencil:}[3\\ 2]\\ \u0026\\ \\\\\\hline \\color{var(--r-Foreground)}[1\\ 2\\ 0] + [1\\ 1\\ 0]\\ \u0026\\ [2\\ 1\\ 0] + [1\\ 1\\ 0]\\\\\\hline \\color{var(--r-Purple)}\\text{Stencil:}[2\\ 3]\\\\\\hline \\color{var(--r-Foreground)}[1\\ 1\\ 0] + [1\\ 2\\ 0]\\ \u0026\\ [1\\ 1\\ 0] + [2\\ 1\\ 0]\\\\\\hline \\end{array} Generalized Montesinos Operation $\\circ$ $\\ $ $= \\color{var(--r-Purple)}([1\\ 2\\ 0] + [1\\ 2\\ 0] + [1\\ 1\\ 0]) \\color{var(--r-Foreground)}\\circ \\color{var(--r-Red)}[1\\ 2]$ Generation We just need to take our lists of Montesinos and rational tangles and glue them together with $\\circ$.\nInto the future Algebraic All possible tangles made from $+$ and $\\vee$\nAlgebraic A vertical sum of two Montesinos tangles. Generation Caudron Trees To generate all possible algebraic tangles, we can generate all possible algebraic expressions on the trivial tangles. Equivalently, all full binary trees with $N$ leaves. Where the tree\u0026rsquo;s internal nodes are labeled with combinations of $\\vee$ and $+$ and leaves are labeled with all combinations of trivial tangles.\nThese binary trees are called Caudron Trees.\nAlain Caudron. Classification des nœuds et des enlacements, volume 4 of Publications Math ́ematiques d\u0026rsquo;Orsay 82 [Mathematical Publications of Orsay 82]. Universit ́e de ParisSud, D ́epartement de Mathe ́matique, Orsay, 1982.\n[3 2 3][3 2 3][3 2 3][3 2 3]++v Non-algebraic/Polygonal 4-valent planar graphs $\\quad$ 4-valent planar graph insertions $6^*\\ *.[1\\ 2\\ 2\\ 3\\ 1].[1\\ 2\\ 2\\ 3\\ 1].[1\\ 2\\ 2\\ 3\\ 1].[1\\ 2\\ 2\\ 3\\ 1].[1\\ 2\\ 2\\ 3\\ 1]$ Generation There exist tables of 4 valent graphs. We can use those with insertions from our list of algebraic tangles to generate all polygonal tangles.\nTooling Design Goals The design for The Tanglenomicon project prioritizes flexibility and extensibility. We want a feature, maybe \u0026ldquo;calculate Jones polynomial,\u0026rdquo; to be runnable in a jupyter notebook or on a university cluster. We\u0026rsquo;re aiming for a \u0026ldquo;write once deploy anywhere\u0026rdquo; design.\nTo that end we\u0026rsquo;ve decoupled functionality wherever feasible, taking a layered approach for system design.\nflowchart LR Runner subgraph \u0026#34;Runnables\u0026#34; Generator Translator Computation end subgraph \u0026#34;Data Wranglers\u0026#34; Notation Storage end Runner --\u0026gt;|Runs| Generator Runner --\u0026gt;|Runs| Computation Runner --\u0026gt;|Runs| Translator Translator --\u0026gt;|Uses| Notation Generator --\u0026gt;|Uses| Notation Computation --\u0026gt;|Uses| Notation Generator --\u0026gt;|Uses| Storage Computation --\u0026gt;|Uses| Storage Translator --\u0026gt;|Uses| Storage Runners A runner is a human/machine interface layer. This abstracts the routines in lower layers for a user to interact with. This could be a CLI, python binding, a Mathematica wrapper, or a web API.\nRunnables Generators\nGenerators create new data. A generator might look like a module to create rational tangles. They may use one or more Computations, Notations, or Translators.\nComputation\nComputations compute a value for a given data. A computation might look like a module for computing a Jones polynomial of a link, or a computing the writhe of a tangle.\nTranslators\nTranslators define a conversion between two Notations. A translator might look like a module for converting from PD notation to Conway notation and back again.\nData Wranglers Notations\nNotations define a notational convention for a link/tangle. They describe a method for converting to and from a string representation of a link/tangle and data structure describing that link/tangle.\nStorage\nA storage module defines a storage interface for the application. The main inter-module type is string and the calling module is responsible for en/decoding the string with a notation module.\nTechnologies ThrowTheSwitch/Unity Simple Unit Testing for C C 3.3k 935 Sources Dror Bar-Natan The Most Important Missing Infrastructure Project in Knot Theory Kauffman, L. H., and S. Lambropoulou. \u0026ldquo;From Tangle Fractions to DNA.\u0026rdquo; In Topology in Molecular Biology, edited by Michail Ilych Monastyrsky, 69-110. Biological and Medical Physics, Biomedical Engineering. Berlin, Heidelberg: Springer Berlin Heidelberg, 2007. https://doi.org/10.1007/978-3-540-49858-2_5. Moon, Hyeyoung, and Isabel K. Darcy. \u0026ldquo;Tangle Equations Involving Montesinos Links.\u0026rdquo; Journal of Knot Theory and Its Ramifications 30, no. 08 (July 2021): 2150060. https://doi.org/10.1142/S0218216521500607. Conway, J.H. \u0026ldquo;An Enumeration of Knots and Links, and Some of Their Algebraic Properties.\u0026rdquo; In Computational Problems in Abstract Algebra, 329-58. Elsevier, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5. Louis H. Kauffman and Sofia Lambropoulou. Classifying and applying rational knots and rational tangles. In DeTurck, editor, Contemporary Mathematics, volume 304, pages 223-259, 2001 Alain Caudron. Classification des nœuds et des enlacements, volume 4 of Publications Math ́ematiques d\u0026rsquo;Orsay 82 [Mathematical Publications of Orsay 82]. Universit ́e de ParisSud, D ́epartement de Mathe ́matique, Orsay, 1982. Robert Glenn Scharein. Interactive topological drawing. ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D. The University of British Columbia (Canada). URL: https://www.knotplot.com/. Jablan, S., \u0026amp; Sazdanović, R. (2007). Linknot. In Series on Knots and Everything. WORLD SCIENTIFIC. https://doi.org/10.1142/6623 Dowker, C. H., \u0026amp; Thistlethwaite, M. B. (1983). Classification of knot projections. In Topology and its Applications (Vol. 16, Issue 1, pp. 19-31). Elsevier BV. https://doi.org/10.1016/0166-8641(83)90004-4 Hoste, J., Thistlethwaite, M., \u0026amp; Weeks, J. (1998). The first 1,701,936 knots. In The Mathematical Intelligencer (Vol. 20, Issue 4, pp. 33-48). Springer Science and Business Media LLC. https://doi.org/10.1007/bf03025227 Burton, B. A. (2020). The Next 350 Million Knots. Schloss Dagstuhl - Leibniz-Zentrum Für Informatik. https://doi.org/10.4230/LIPICS.SOCG.2020.25 C. Livingston and A. H. Moore, KnotInfo: Table of Knot Invariants, knotinfo.math.indiana.edu, today\u0026rsquo;s date (eg. August 24, 2023). Schubert, Horst. \u0026ldquo;Knoten mit zwei Brücken..\u0026rdquo; Mathematische Zeitschrift 65 (1956): 133-170. http://eudml.org/doc/169591. Jos ́e M. Montesinos. Seifert manifolds that are ramified two-sheeted cyclic coverings. Bol. Soc. Mat. Mexicana (2), 18:1-32, 1973. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html ","date":"15 November 2023","externalUrl":null,"permalink":"/speaking/research/dearborn_f23/","section":"Slides","summary":"Talk given at the U of M - Dearborn Colloquium on 11/15/23.","title":"Dearborn Colloquium Talk Fall 2023","type":"slides"},{"content":"","date":"10 November 2023","externalUrl":null,"permalink":"/speaking/posters/9mwagt/","section":"Slides","summary":"","title":"9th Mexican Workshop on Applied Geometry and Topology ","type":"posters"},{"content":" Joecstarr/knotplot_qr Lua 1 1 $$\\ $$\nA knotplot qr code generator This is an addon script for Rob Scharein\u0026rsquo;s great knot drawing tool knotplot. Rob has recently included the Lua scripting engine into knotplot allowing for some fairly advanced interactions with knotplot.\nThis script uses the knotplot\u0026rsquo;s celtic knot drawing tools to draw \u0026ldquo;LR\u0026rdquo; codes (qr codes but made of links). Some examples:\nHow to use The script runs entirely in knotplot so you need to purchase and install knotplot. The script currently depends on a beta version of knotplot which will be wildly available soon.\nGit pull Start by pulling this repo into your knotplot workspace.\ngit pull https://github.com/Joecstarr/knotplot_qr.git Git update submodule The script uses the luaqrcode library. The library is referenced as a git submodule which can be pulled by running\ngit submodule update --init --recursive Run script Run the script in knotplot by using the lua run command. The first argument is interpreted as the string to convert to QR code.\nlua run knotplot_qr/knotplot_qr.lua \u0026lt;string\u0026gt; Post processing The output of the script is an postscript (.eps) file. This can be turned into an image by using inkscape or imagemagick.\n","date":"16 October 2023","externalUrl":null,"permalink":"/resources/tools/text_knots/","section":"Resources","summary":"This script uses the knotplot\u0026rsquo;s celtic knot drawing tools to draw \u0026lsquo;LR\u0026rsquo; codes (qr codes but made of links).","title":"A Knotplot QR Code Generator","type":"resources"},{"content":" Joecstarr/knotplot_text Lua 1 0 $$\\ $$\nA knotplot celtic text generator Under construction!\n","date":"16 October 2023","externalUrl":null,"permalink":"/resources/tools/qr_code_knots/","section":"Resources","summary":"This script uses the knotplot\u0026rsquo;s celtic knot drawing tools to draw text.","title":"A Knotplot Text Generator","type":"resources"},{"content":"","date":"16 October 2023","externalUrl":null,"permalink":"/tags/tool/","section":"Tags","summary":"","title":"Tool","type":"tags"},{"content":"","date":"16 October 2023","externalUrl":null,"permalink":"/tags/visualizer/","section":"Tags","summary":"","title":"Visualizer","type":"tags"},{"content":" $\\newcommand{\\N}{\\mathbb{N}} \\newcommand{\\Z}{\\mathbb{Z}} \\newcommand{\\Q}{\\mathbb{Q}} \\newcommand{\\R}{\\mathbb{R}} \\newcommand{\\LP}{\\left(} \\newcommand{\\RP}{\\right)} \\newcommand{\\LS}{\\left\\lbrace} \\newcommand{\\RS}{\\right\\rbrace} \\newcommand{\\LA}{\\left\\langle} \\newcommand{\\RA}{\\right\\rangle} \\newcommand{\\LB}{\\left[} \\newcommand{\\RB}{\\right]} \\newcommand{\\MM}{\\ \\middle|\\ } \\newcommand{\\exp}{\\text{exp}} \\newcommand{\\abs}[1]{\\left\\vert#1\\right\\vert} \\newcommand{\\msr}[1]{m\\left(#1\\right)} \\newcommand{\\inv}[1]{#1^{-1}} \\newcommand{\\bkt}[1]{\\LA \\img{#1}\\RA} \\require{color}$ The Tanglenomicon Zachary Bryhtan, Nicholas Connolly, Isabel Darcy, Ethan Rooke, Joseph Starr* Mathematics Department at The University of Iowa Knots \u0026ldquo;A knot is a smooth embedding of a circle $S^1$ into Euclidean 3-dimensional space $\\R^3$ (or the 3-dimensional sphere $S^3$ ).\u0026rdquo;\n$\\quad$ $\\quad$ $\\quad$ Jablan, S., \u0026amp; Sazdanović, R. (2007). Linknot. In Series on Knots and Everything. WORLD SCIENTIFIC. https://doi.org/10.1142/6623\nhttps://www.knotplot.com/\nKnot Tables Lord Kelvin\u0026rsquo;s vortex theory of the atom. Atoms are knotted vortices in the æther. 1860\u0026rsquo;s Tait computes knots up to 7 crossing 15 knots 1870\u0026rsquo;s Tait, Kirkman, and Little compute knots up to 10 crossing Takes about 25 years 250 knots 1960\u0026rsquo;s Conway computes knots up to 11 crossings \u0026ldquo;A few hours\u0026rdquo; 802 knots 1980\u0026rsquo;s Dowker and Thistlethwaite compute up to 13 crossings First using a computer 12,966 1990\u0026rsquo;s Hoste, Thistlethwaite, and Weeks compute up to 16 crossings Computer runtime on the order of weeks 1,701,936 2020\u0026rsquo;s Burton computes up to 19 crossings 350 Million KnotInfo Conway How did Conway compute 25 years of work in \"a few hours\"? Tangles \u0026ldquo;We define a tangle as a portion of a knot diagram from which there emerge just 4 arcs pointing in the compass directions NW, NE, SW, SE.\u0026rdquo;\nConway, J.H. \u0026ldquo;An Enumeration of Knots and Links, and Some of Their Algebraic Properties.\u0026rdquo; In Computational Problems in Abstract Algebra, 329-58. Elsevier, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5\n$\\quad$ $\\quad$ NWNESWSE $\\quad$ $\\quad$ $\\quad$ Basic Operations Operation $+$ $+$ $=$ $=$ $=$ $2$ Operation $\\vee$ $\\vee$ $=$ $=$ $=$ $\\frac{1}{2}$ The Tanglenomicon Building up $\\ $ $\\ $ $\\ $ $\\ $ Where we are Rational Tangles $\\ $ $\\begin{aligned}\\to\u0026\\ \\LP 3 \\vee \\frac{1}{2}\\RP + 2\\\\\u0026\\\\ \\to\u0026\\ [3\\ 2\\ 2]\\end{aligned}$ Where we\u0026rsquo;re going Montesinos $+$ $=$ $$\\ =\\ $$ $$=[3\\ 2\\ 2] + [3\\ 2\\ 2]$$ Generalized Montesinos \u0026lt;img class=\u0026quot;centerImg\u0026quot; style=\u0026quot; width:35rem; \u0026quot; src=\u0026quot; /presentations/lightning/GenMont.svg\u0026quot;/\u0026gt; Operation $\\circ$ $\\ $ $= \\color{var(--r-Purple)}([1\\ 2\\ 0] + [1\\ 2\\ 0] + [1\\ 1\\ 0]) \\color{var(--r-Foreground)}\\circ \\color{var(--r-Red)}[1\\ 2]$ Into the future Algebraic All possible tangles made from $+$ and $\\vee$\nAlgebraic A vertical sum of two Montesinos tangles. Non-algebraic/Polygonal 4-valent planar graphs \u0026lt;img class=\u0026quot;centerImg\u0026quot; style=\u0026quot; width:15rem; \u0026quot; src=\u0026quot; /presentations/general/1star.svg\u0026quot;/\u0026gt; \u0026lt;img class=\u0026quot;centerImg\u0026quot; style=\u0026quot; width:30rem; \u0026quot; src=\u0026quot; /presentations/general/6star.svg\u0026quot;/\u0026gt; 4-valent planar graph insertions Some useful links Personal Site joe-starr.com\nMGB mathgradboard.com\nKnotPlot knotplot.com\nKnot Info knotinfo.math.indiana.edu\nIllustrating Mathematics Discord discord.gg/jedHjNgZn\nSeminars Topology Reading Seminar T 2pm-3pm 221 MLH Knot, Tangles, and Computers Th 11:30am - 12:30pm Topology Research Seminar Th 2pm-3pm 221 MLH Topological Data Visualization F 3:30pm - 4:30pm B13 MLH Questions? Sources Dror Bar-Natan The Most Important Missing Infrastructure Project in Knot Theory Kauffman, L. H., and S. Lambropoulou. \u0026ldquo;From Tangle Fractions to DNA.\u0026rdquo; In Topology in Molecular Biology, edited by Michail Ilych Monastyrsky, 69-110. Biological and Medical Physics, Biomedical Engineering. Berlin, Heidelberg: Springer Berlin Heidelberg, 2007. https://doi.org/10.1007/978-3-540-49858-2_5. Moon, Hyeyoung, and Isabel K. Darcy. \u0026ldquo;Tangle Equations Involving Montesinos Links.\u0026rdquo; Journal of Knot Theory and Its Ramifications 30, no. 08 (July 2021): 2150060. https://doi.org/10.1142/S0218216521500607. Conway, J.H. \u0026ldquo;An Enumeration of Knots and Links, and Some of Their Algebraic Properties.\u0026rdquo; In Computational Problems in Abstract Algebra, 329-58. Elsevier, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5. Louis H. Kauffman and Sofia Lambropoulou. Classifying and applying rational knots and rational tangles. In DeTurck, editor, Contemporary Mathematics, volume 304, pages 223-259, 2001 Alain Caudron. Classification des nœuds et des enlacements, volume 4 of Publications Math ́ematiques d\u0026rsquo;Orsay 82 [Mathematical Publications of Orsay 82]. Universit ́e de ParisSud, D ́epartement de Mathe ́matique, Orsay, 1982. Robert Glenn Scharein. Interactive topological drawing. ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D. The University of British Columbia (Canada). URL: https://www.knotplot.com/. Jablan, S., \u0026amp; Sazdanović, R. (2007). Linknot. In Series on Knots and Everything. WORLD SCIENTIFIC. https://doi.org/10.1142/6623 Dowker, C. H., \u0026amp; Thistlethwaite, M. B. (1983). Classification of knot projections. In Topology and its Applications (Vol. 16, Issue 1, pp. 19-31). Elsevier BV. https://doi.org/10.1016/0166-8641(83)90004-4 Hoste, J., Thistlethwaite, M., \u0026amp; Weeks, J. (1998). The first 1,701,936 knots. In The Mathematical Intelligencer (Vol. 20, Issue 4, pp. 33-48). Springer Science and Business Media LLC. https://doi.org/10.1007/bf03025227 Burton, B. A. (2020). The Next 350 Million Knots. Schloss Dagstuhl - Leibniz-Zentrum Für Informatik. https://doi.org/10.4230/LIPICS.SOCG.2020.25 C. Livingston and A. H. Moore, KnotInfo: Table of Knot Invariants, knotinfo.math.indiana.edu, today\u0026rsquo;s date (eg. August 24, 2023). ","date":"22 August 2023","externalUrl":null,"permalink":"/speaking/research/mathday23/","section":"Slides","summary":"Talk given at the Fall23 University of Iowa Math Day","title":"University of Iowa Fall23 Math Day","type":"slides"},{"content":" $\\newcommand{\\N}{\\mathbb{N}} \\newcommand{\\Z}{\\mathbb{Z}} \\newcommand{\\Q}{\\mathbb{Q}} \\newcommand{\\R}{\\mathbb{R}} \\newcommand{\\LP}{\\left(} \\newcommand{\\RP}{\\right)} \\newcommand{\\LS}{\\left\\lbrace} \\newcommand{\\RS}{\\right\\rbrace} \\newcommand{\\LA}{\\left\\langle} \\newcommand{\\RA}{\\right\\rangle} \\newcommand{\\LB}{\\left[} \\newcommand{\\RB}{\\right]} \\newcommand{\\MM}{\\ \\middle|\\ } \\newcommand{\\exp}{\\text{exp}} \\newcommand{\\abs}[1]{\\left\\vert#1\\right\\vert} \\newcommand{\\msr}[1]{m\\left(#1\\right)} \\newcommand{\\inv}[1]{#1^{-1}} \\newcommand{\\bkt}[1]{\\LA \\img{#1}\\RA} \\require{color}$ The Tanglenomicon Zachary Bryhtan, Nicholas Connolly, Isabel Darcy, Ethan Rooke, Joseph Starr* Mathematics Department at The University of Iowa Tangle tabulation \u0026ldquo;The Most Important Missing Infrastructure Project in Knot Theory\u0026rdquo; -Dr. Dror Bar-Natan [2012]\nClassified Rational Tangles $$[3\\ 2\\ 2]$$\n$$([1\\ 2\\ 0] + [1\\ 2\\ 0] + [1\\ 1\\ 0]) \\circ (2,2)$$ $$\\rightarrow$$ Montesinos Tangles $$[3 0 ] + [2 1 0] + [2 2 0]$$\n$$([1\\ 2\\ 0] + [1\\ 2\\ 0] + [1\\ 1\\ 0]) \\circ (2,2)$$ $$\\rightarrow$$ Generalized Montesinos Tangles $$([1\\ 2\\ 0] + [1\\ 2\\ 0] + [1\\ 1\\ 0]) \\circ (2,2)$$\n$$([1\\ 2\\ 0] + [1\\ 2\\ 0] + [1\\ 1\\ 0]) \\circ (2,2)$$ Not Classified Algebraic Tangles $$([3\\ 2\\ 3] + [3\\ 2\\ 3]) \\vee ([3\\ 2\\ 3] + [3\\ 2\\ 3] )$$ $$6^\\ast\\ *.[3\\ 2\\ 3\\ 1].[3\\ 2\\ 3\\ 1].[3\\ 2\\ 3\\ 1].[3\\ 2\\ 3\\ 1].[3\\ 2\\ 3\\ 1]$$ $$\\rightarrow$$ Non-Algebraic Tangles \u0026lt;path d=\u0026quot;M 263.32314,556.50083 C 235.70855,506.14121 243.53544,430.34458 293.93443,407.56097\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;round\u0026quot; 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fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path90\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 635.14077,317.11617 -3.615,-3.615 -1.856,-1.856\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path92\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 597.89977,277.22917 c 1.853,1.853 3.618,3.618 5.471,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path94\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 597.89977,277.22917 c 1.853,1.853 3.618,3.618 5.471,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path96\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 603.19377,282.52317 c 1.854,1.85401 3.619,3.619 5.472,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path98\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 603.19377,282.52317 c 1.854,1.85401 3.619,3.619 5.472,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path100\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 560.83477,298.40717 c 1.853,1.853 3.618,3.618 5.471,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path102\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 560.83477,298.40717 c 1.853,1.853 3.618,3.618 5.471,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path104\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 637.69877,324.23717 c 0,-3.091 -0.881,-5.444 -2.734,-7.297\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path106\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 637.69877,324.23717 c 0,-3.091 -0.881,-5.444 -2.734,-7.297\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path108\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 576.89577,346.06417 c -1.854,1.853 -1.854,3.618 0,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path110\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 576.89577,346.06417 c -1.854,1.853 -1.854,3.618 0,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path112\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 629.84377,311.82218 -1.805,-1.80501 -3.666,-3.66699\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path114\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 582.01277,340.76717 c 1.854,1.85301 3.619,3.618 5.472,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path116\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 582.01277,340.76717 c 1.854,1.85301 3.619,3.618 5.472,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path118\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 568.68677,315.66118 c 3.707,0.925 6.355,2.25099 8.209,4.10399\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path120\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 568.68677,315.66118 c 3.707,0.925 6.355,2.25099 8.209,4.10399\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path122\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 555.71477,271.93218 c -1.854,1.853 -1.854,3.618 0,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path124\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 555.71477,271.93218 c -1.854,1.853 -1.854,3.618 0,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path126\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 582.01277,324.88317 c 1.854,1.853 3.619,3.618 5.472,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path128\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 582.01277,324.88317 c 1.854,1.853 3.619,3.618 5.472,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path130\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 576.89577,330.17717 c -1.854,1.854 -1.854,3.618 0,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path132\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 576.89577,330.17717 c -1.854,1.854 -1.854,3.618 0,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path134\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 560.83477,266.63718 c 1.853,1.853 3.618,3.61799 5.471,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path136\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 560.83477,266.63718 c 1.853,1.853 3.618,3.61799 5.471,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path138\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 598.07677,287.81817 c -1.854,1.853 -3.621,1.853 -5.474,0\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path140\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 598.07677,287.81817 c -1.854,1.853 -3.621,1.853 -5.474,0\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path142\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 555.71477,287.81817 c -1.854,1.853 -1.854,3.618 0,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path144\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 552.97977,254.22517 c 0,3.088 0.881,5.441 2.735,7.29401\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path146\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 552.97977,254.22517 c 0,3.088 0.881,5.441 2.735,7.29401\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path148\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 587.48477,282.70017 c -1.853,-1.853 -3.618,-3.618 -5.472,-5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path150\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 587.48477,282.70017 c -1.853,-1.853 -3.618,-3.618 -5.472,-5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path152\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 592.60277,277.40318 c 1.853,-1.853 3.62,-1.853 5.474,0\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path154\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 592.60277,277.40318 c 1.853,-1.853 3.62,-1.853 5.474,0\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path156\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 560.83477,282.52317 c 1.853,1.85401 3.618,3.619 5.471,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path158\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 560.83477,282.52317 c 1.853,1.85401 3.618,3.619 5.471,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path160\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 552.97977,308.99617 c 0,1.854 1.765,3.179 5.472,4.105\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path162\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 552.97977,308.99617 c 0,1.854 1.765,3.179 5.472,4.105\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path164\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 595.51577,315.66118 c -3.706,0.925 -6.355,2.25099 -8.208,4.10399\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path166\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 595.51577,315.66118 c -3.706,0.925 -6.355,2.25099 -8.208,4.10399\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path168\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 596.44177,364.68518 c 4.634,0 9.048,0 13.679,0\u0026quot; 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fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path200\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 617.88777,282.19218 c -0.928,-5.096 -2.251,-7.52301 -4.104,-7.52301\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path202\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 617.88777,282.19218 c -0.928,-5.096 -2.251,-7.52301 -4.104,-7.52301\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; 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fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path220\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 582.18977,356.65317 c -1.853,1.853 -3.618,3.618 -5.471,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path222\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 582.18977,356.65317 c -1.853,1.853 -3.618,3.618 -5.471,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path224\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 603.37077,282.52317 c -1.853,1.85401 -3.618,3.619 -5.471,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path226\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 603.37077,282.52317 c -1.853,1.85401 -3.618,3.619 -5.471,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path228\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 582.18977,340.76717 c -1.853,1.85301 -3.618,3.618 -5.471,5.471\u0026quot; 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stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path234\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 617.37377,357.43217 c 0.616,-5.251 1.204,-10.252 1.823,-15.503\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path236\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 587.30777,330.17717 c 1.853,1.854 1.853,3.618 0,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path238\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 587.30777,330.17717 c 1.853,1.854 1.853,3.618 0,5.472\u0026quot; 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fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path270\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 582.01277,287.99518 c 1.854,-1.85301 3.619,-3.618 5.472,-5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path272\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 582.01277,287.99518 c 1.854,-1.85301 3.619,-3.618 5.472,-5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path274\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 566.30577,261.34217 c -1.853,1.85401 -3.618,3.61901 -5.471,5.47201\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path276\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 566.30577,261.34217 c -1.853,1.85401 -3.618,3.61901 -5.471,5.47201\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path278\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 587.48477,351.35818 c -1.853,1.85399 -3.618,3.619 -5.472,5.47199\u0026quot; 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stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path314\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 561.00877,282.52317 c -1.853,1.85401 -3.618,3.619 -5.471,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path316\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 561.00877,282.52317 c -1.853,1.85401 -3.618,3.619 -5.471,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path318\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 587.48477,319.58817 c -1.853,1.853 -3.618,3.618 -5.472,5.472\u0026quot; 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fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path330\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 566.30577,293.11317 c -1.853,1.85301 -3.618,3.618 -5.471,5.47101\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path332\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 566.30577,293.11317 c -1.853,1.85301 -3.618,3.618 -5.471,5.47101\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; 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id=\u0026quot;path422\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 513.90942,500.5434 c -1.853,1.853 -1.853,3.618 0,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path424\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 540.20842,553.4944 c 1.853,1.853 3.618,3.618 5.472,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path426\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 540.20842,553.4944 c 1.853,1.853 3.618,3.618 5.472,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; 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id=\u0026quot;path432\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 519.03042,495.2484 c 1.853,1.853 3.618,3.618 5.471,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path434\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 519.03042,495.2484 c 1.853,1.853 3.618,3.618 5.471,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path436\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 556.27142,516.4294 c -1.853,1.853 -3.62,1.853 -5.474,0\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path438\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 556.27142,516.4294 c -1.853,1.853 -3.62,1.853 -5.474,0\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path440\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 513.90942,516.4294 c -1.853,1.853 -1.853,3.618 0,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path442\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 511.17542,482.8364 c 0,3.088 0.881,5.441 2.734,7.294\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path444\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 511.17542,482.8364 c 0,3.088 0.881,5.441 2.734,7.294\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path446\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 545.68042,511.3114 c -1.854,-1.853 -3.619,-3.618 -5.472,-5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path448\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 545.68042,511.3114 c -1.854,-1.853 -3.619,-3.618 -5.472,-5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path450\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 550.79742,506.0144 c 1.854,-1.853 3.621,-1.853 5.474,0\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path452\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 550.79742,506.0144 c 1.854,-1.853 3.621,-1.853 5.474,0\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path454\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 519.03042,511.1344 c 1.853,1.854 3.618,3.619 5.471,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path456\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 519.03042,511.1344 c 1.853,1.854 3.618,3.619 5.471,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path458\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 511.17542,537.6074 c 0,1.854 1.765,3.179 5.471,4.105\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path460\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 511.17542,537.6074 c 0,1.854 1.765,3.179 5.471,4.105\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path462\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 553.71142,544.2724 c -3.706,0.925 -6.355,2.251 -8.208,4.104\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path464\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 553.71142,544.2724 c -3.706,0.925 -6.355,2.251 -8.208,4.104\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path466\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 554.63642,593.2964 c 4.635,0 9.048,0 13.68,0\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path468\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 554.63642,593.2964 c 4.635,0 9.048,0 13.68,0\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path470\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 566.68442,516.4294 c 1.853,1.853 2.734,4.499 2.734,8.206\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path472\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 566.68442,516.4294 c 1.853,1.853 2.734,4.499 2.734,8.206\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path474\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 567.87442,593.2964 c 4.635,0 7.134,-2.503 7.753,-7.753\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path476\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 567.87442,593.2964 c 4.635,0 7.134,-2.503 7.753,-7.753\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path478\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 569.41842,534.8734 c 0,3.707 -1.765,5.913 -5.471,6.839\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path480\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 569.41842,534.8734 c 0,3.707 -1.765,5.913 -5.471,6.839\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path482\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 582.74442,535.1384 c -1.853,-1.853 -3.176,-5.162 -4.101,-10.26\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path484\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 582.74442,535.1384 c -1.853,-1.853 -3.176,-5.162 -4.101,-10.26\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path486\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 545.50342,511.3114 c 1.853,-1.853 3.618,-3.618 5.471,-5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path488\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 545.50342,511.3114 c 1.853,-1.853 3.618,-3.618 5.471,-5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path490\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 578.72942,525.3644 c -0.926,-5.096 -1.81,-9.951 -2.735,-15.047\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path492\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 578.72942,525.3644 c -0.926,-5.096 -1.81,-9.951 -2.735,-15.047\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path494\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 540.38542,553.4944 c -1.853,1.853 -3.618,3.618 -5.471,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path496\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 540.38542,553.4944 c -1.853,1.853 -3.618,3.618 -5.471,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path498\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 576.08342,510.8034 c -0.928,-5.096 -2.251,-7.523 -4.105,-7.523\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path500\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 576.08342,510.8034 c -0.928,-5.096 -2.251,-7.523 -4.105,-7.523\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; 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fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path518\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 540.38542,585.2644 c -1.853,1.853 -3.618,3.618 -5.471,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path520\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 540.38542,585.2644 c -1.853,1.853 -3.618,3.618 -5.471,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; 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stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path552\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 540.38542,506.0144 c -1.853,-1.853 -3.618,-5.383 -5.471,-10.943\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path554\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 540.38542,506.0144 c -1.853,-1.853 -3.618,-5.383 -5.471,-10.943\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path556\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 535.09042,495.6024 c -1.853,-5.56 -3.618,-8.209 -5.471,-8.209\u0026quot; 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stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path572\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 524.50142,489.9534 c -1.853,1.854 -3.618,3.619 -5.471,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path574\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 524.50142,489.9534 c -1.853,1.854 -3.618,3.619 -5.471,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path576\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 545.68042,579.9694 c -1.854,1.854 -3.619,3.619 -5.472,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path578\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 545.68042,579.9694 c -1.854,1.854 -3.619,3.619 -5.472,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path580\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 519.20442,495.2484 c -1.853,1.853 -3.618,3.618 -5.471,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path582\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 519.20442,495.2484 c -1.853,1.853 -3.618,3.618 -5.471,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path584\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 545.68042,564.0834 c -1.854,1.853 -3.619,3.618 -5.472,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path586\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 545.68042,564.0834 c -1.854,1.853 -3.619,3.618 -5.472,5.471\u0026quot; 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id=\u0026quot;path790\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 371.9712,346.90103 c -0.925,-5.096 -1.809,-9.951 -2.734,-15.047\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path792\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 333.6282,375.03103 c -1.854,1.853 -3.619,3.618 -5.472,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path794\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 333.6282,375.03103 c -1.854,1.853 -3.619,3.618 -5.472,5.471\u0026quot; fill=\u0026quot;none\u0026quot; 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stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path910\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 312.4472,332.67203 c -1.854,1.853 -3.619,3.618 -5.472,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path912\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 312.4472,332.67203 c -1.854,1.853 -3.619,3.618 -5.472,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path914\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 338.9222,369.73703 c -1.853,1.853 -3.618,3.618 -5.471,5.471\u0026quot; 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id=\u0026quot;path1276\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 268.88754,562.10875 c 1.853,1.853 3.618,3.618 5.471,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path1278\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 268.88754,562.10875 c 1.853,1.853 3.618,3.618 5.471,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path1280\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 348.49054,617.88275 c -1.853,-1.853 -3.618,-3.618 -5.471,-5.471\u0026quot; fill=\u0026quot;none\u0026quot; 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stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path1286\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 311.24954,577.99575 c 1.85301,1.853 3.618,3.618 5.47101,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path1288\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 316.54354,583.28975 c 1.854,1.853 3.61901,3.618 5.472,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path1290\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 316.54354,583.28975 c 1.854,1.853 3.61901,3.618 5.472,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path1292\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 274.18455,599.17375 c 1.853,1.853 3.61799,3.618 5.471,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path1294\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 274.18455,599.17375 c 1.853,1.853 3.61799,3.618 5.471,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; 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fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path1302\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 290.24554,646.82975 c -1.854,1.854 -1.854,3.619 0,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path1304\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 343.19355,612.58775 -1.805,-1.805 -3.666,-3.666\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path1306\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 295.36254,641.53275 c 1.854,1.853 3.618,3.618 5.472,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path1308\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 295.36254,641.53275 c 1.854,1.853 3.618,3.618 5.472,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path1310\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 282.03654,616.42675 c 3.707,0.926 6.355,2.251 8.209,4.105\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path1312\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 282.03654,616.42675 c 3.707,0.926 6.355,2.251 8.209,4.105\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path1314\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 269.06454,572.69775 c -1.85399,1.853 -1.85399,3.618 0,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path1316\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 269.06454,572.69775 c -1.85399,1.853 -1.85399,3.618 0,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path1318\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 295.36254,625.64875 c 1.854,1.854 3.618,3.619 5.472,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path1320\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 295.36254,625.64875 c 1.854,1.854 3.618,3.619 5.472,5.472\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path1322\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 290.24554,630.94375 c -1.854,1.853 -1.854,3.618 0,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path1324\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 290.24554,630.94375 c -1.854,1.853 -1.854,3.618 0,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path1326\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 274.18455,567.40375 c 1.853,1.853 3.61799,3.618 5.471,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path1328\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 274.18455,567.40375 c 1.853,1.853 3.61799,3.618 5.471,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path1330\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 311.42655,588.58475 c -1.854,1.853 -3.621,1.853 -5.474,0\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path1332\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 311.42655,588.58475 c -1.854,1.853 -3.621,1.853 -5.474,0\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path1334\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 269.06454,588.58475 c -1.85399,1.853 -1.85399,3.618 0,5.471\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path1336\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 266.32954,554.99175 c 0,3.087 0.88101,5.441 2.735,7.294\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-selection\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;7\u0026quot; id=\u0026quot;path1338\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 266.32954,554.99175 c 0,3.087 0.88101,5.441 2.735,7.294\u0026quot; fill=\u0026quot;none\u0026quot; opacity=\u0026quot;1\u0026quot; class=\u0026quot;stroke-orange\u0026quot; stroke=\u0026quot;#ffb86c\u0026quot; stroke-linecap=\u0026quot;round\u0026quot; stroke-linejoin=\u0026quot;miter\u0026quot; stroke-width=\u0026quot;4\u0026quot; id=\u0026quot;path1340\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 300.83454,583.46675 c -1.854,-1.853 -3.618,-3.618 -5.472,-5.471\u0026quot; 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class=\u0026quot;stroke-highlight fill-highlight\u0026quot; stroke=\u0026quot;#6272a4\u0026quot; fill=\u0026quot;#6272a4\u0026quot; inkscape:label=\u0026quot;path21850\u0026quot; style=\u0026quot;stroke-width:4.43177\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 785.36816,118.8878 c 0,-42.065 34.101,-76.166 76.166,-76.166 42.065,0 76.166,34.101 76.166,76.166 0,42.065 -34.101,76.165 -76.166,76.165 -42.065,0 -76.166,-34.1 -76.166,-76.165 z\u0026quot; class=\u0026quot;fill-selection stroke-selection\u0026quot; fill=\u0026quot;#44475a\u0026quot; fill-rule=\u0026quot;nonzero\u0026quot; opacity=\u0026quot;1\u0026quot; stroke=\u0026quot;#44475a\u0026quot; stroke-linecap=\u0026quot;butt\u0026quot; stroke-linejoin=\u0026quot;round\u0026quot; stroke-width=\u0026quot;0.1\u0026quot; id=\u0026quot;path51\u0026quot; /\u0026gt; \u0026lt;path d=\u0026quot;m 883.79904,118.88731 32.16038,16.69865 -12.98785,22.26488 -30.30497,-19.17253 1.85541,35.87118 h -25.97569 l 1.85541,-35.87118 -30.30497,19.17253 -12.98785,-22.26488 31.54191,-16.69865 -31.54191,-16.69866 12.98785,-22.26488 30.30497,19.17253 -1.85541,-35.87118 h 25.97569 l -1.85541,35.87118 30.30497,-19.17253 12.98785,22.26488 z\u0026quot; id=\u0026quot;text2825\u0026quot; class=\u0026quot;fill-orange\u0026quot; style=\u0026quot;font-size:158.328px;-inkscape-font-specification:'sans-serif, Normal';fill:#ffb86c;fill-opacity:1;stroke-width:13.194\u0026quot; aria-label=\u0026quot;✱\u0026quot; /\u0026gt; $$6^\\ast\\ *.[3\\ 2\\ 3\\ 1].[3\\ 2\\ 3\\ 1].[3\\ 2\\ 3\\ 1].[3\\ 2\\ 3\\ 1].[3\\ 2\\ 3\\ 1]$$ $$6^\\ast\\ *.[3\\ 2\\ 3\\ 1].[3\\ 2\\ 3\\ 1].[3\\ 2\\ 3\\ 1].[3\\ 2\\ 3\\ 1].[3\\ 2\\ 3\\ 1]$$ Sources Dror Bar-Natan The Most Important Missing Infrastructure Project in Knot Theory Kauffman, L. H., and S. Lambropoulou. \u0026ldquo;From Tangle Fractions to DNA.\u0026rdquo; In Topology in Molecular Biology, edited by Michail Ilych Monastyrsky, 69-110. Biological and Medical Physics, Biomedical Engineering. Berlin, Heidelberg: Springer Berlin Heidelberg, 2007. https://doi.org/10.1007/978-3-540-49858-2_5. Moon, Hyeyoung, and Isabel K. Darcy. \u0026ldquo;Tangle Equations Involving Montesinos Links.\u0026rdquo; Journal of Knot Theory and Its Ramifications 30, no. 08 (July 2021): 2150060. https://doi.org/10.1142/S0218216521500607. Conway, J.H. \u0026ldquo;An Enumeration of Knots and Links, and Some of Their Algebraic Properties.\u0026rdquo; In Computational Problems in Abstract Algebra, 329-58. Elsevier, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5. Louis H. Kauffman and Sofia Lambropoulou. Classifying and applying rational knots and rational tangles. In DeTurck, editor, Contemporary Mathematics, volume 304, pages 223-259, 2001 Alain Caudron. Classification des nœuds et des enlacements, volume 4 of Publications Math ́ematiques d\u0026rsquo;Orsay 82 [Mathematical Publications of Orsay 82]. Universit ́e de ParisSud, D ́epartement de Mathe ́matique, Orsay, 1982. Robert Glenn Scharein. Interactive topological drawing. ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D. The University of British Columbia (Canada). URL: https://www.knotplot.com/. Jablan, S., \u0026amp; Sazdanović, R. (2007). Linknot. In Series on Knots and Everything. WORLD SCIENTIFIC. https://doi.org/10.1142/6623 Dowker, C. H., \u0026amp; Thistlethwaite, M. B. (1983). Classification of knot projections. In Topology and its Applications (Vol. 16, Issue 1, pp. 19-31). Elsevier BV. https://doi.org/10.1016/0166-8641(83)90004-4 Hoste, J., Thistlethwaite, M., \u0026amp; Weeks, J. (1998). The first 1,701,936 knots. In The Mathematical Intelligencer (Vol. 20, Issue 4, pp. 33-48). Springer Science and Business Media LLC. https://doi.org/10.1007/bf03025227 Burton, B. A. (2020). The Next 350 Million Knots. Schloss Dagstuhl - Leibniz-Zentrum Für Informatik. https://doi.org/10.4230/LIPICS.SOCG.2020.25 C. Livingston and A. H. Moore, KnotInfo: Table of Knot Invariants, knotinfo.math.indiana.edu, today\u0026rsquo;s date (eg. August 24, 2023). Schubert, Horst. \u0026ldquo;Knoten mit zwei Brücken..\u0026rdquo; Mathematische Zeitschrift 65 (1956): 133-170. http://eudml.org/doc/169591. Jos ́e M. Montesinos. Seifert manifolds that are ramified two-sheeted cyclic coverings. Bol. Soc. Mat. Mexicana (2), 18:1-32, 1973. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html Visit: https://joe-starr.com\n","date":"10 May 2023","externalUrl":null,"permalink":"/speaking/research/lightning/","section":"Slides","summary":"Lightning talk for Tangled in Knot Theory","title":"Tangled in Knot Theory Summer 2023","type":"slides"},{"content":" $\\newcommand{\\N}{\\mathbb{N}} \\newcommand{\\Z}{\\mathbb{Z}} \\newcommand{\\Q}{\\mathbb{Q}} \\newcommand{\\R}{\\mathbb{R}} \\newcommand{\\LP}{\\left(} \\newcommand{\\RP}{\\right)} \\newcommand{\\LS}{\\left\\lbrace} \\newcommand{\\RS}{\\right\\rbrace} \\newcommand{\\LA}{\\left\\langle} \\newcommand{\\RA}{\\right\\rangle} \\newcommand{\\LB}{\\left[} \\newcommand{\\RB}{\\right]} \\newcommand{\\MM}{\\ \\middle|\\ } \\newcommand{\\exp}{\\text{exp}} \\newcommand{\\abs}[1]{\\left\\vert#1\\right\\vert} \\newcommand{\\msr}[1]{m\\left(#1\\right)} \\newcommand{\\inv}[1]{#1^{-1}} \\newcommand{\\bkt}[1]{\\LA \\img{#1}\\RA} \\require{color}$ Alexander Polynomial Seifert surfaces Examples \u0026lt;img class=\u0026quot;centerImg\u0026quot; style=\u0026quot; max-width:500px; \u0026quot; src=\u0026quot; /presentations/Alex_Poly/bands/Band.svg\u0026quot;/\u0026gt; \u0026lt;img class=\u0026quot;centerImg\u0026quot; style=\u0026quot; max-width:500px; \u0026quot; src=\u0026quot; /presentations/Alex_Poly/alg/Alg_7.svg\u0026quot;/\u0026gt; Definition A Seifert surface for an oriented link in $S^3$ is a compact connected oriented surface smoothly embedded in $S^3$ with oriented boundary equal to the link.\nExistence Existence can be shown by an algorithm to construct a Seifert surface from a given link projection.\nRemove crossings Close curves Collection of disks Attaching bands \u0026lt;img class=\u0026quot;centerImg\u0026quot; style=\u0026quot; max-width:500px; \u0026quot; src=\u0026quot; /presentations/Alex_Poly/alg/Alg_4.svg\u0026quot;/\u0026gt; \u0026lt;img class=\u0026quot;centerImg\u0026quot; style=\u0026quot; max-width:500px; \u0026quot; src=\u0026quot; /presentations/Alex_Poly/alg/Alg_5.svg\u0026quot;/\u0026gt; Construction Remove crossings. Connect strands following orientation without creating new crossings. Fill interior of resulting disks. Connect disks with \u0026ldquo;twists\u0026rdquo; matching crossing orientation. SeifertView of $6_{2}$ Bands \u0026lt;img class=\u0026quot;centerImg\u0026quot; style=\u0026quot; max-width:500px; \u0026quot; src=\u0026quot; /presentations/Alex_Poly/alg/Alg_7.svg\u0026quot;/\u0026gt; \u0026lt;p\u0026gt;$$\\to$$\u0026lt;/p\u0026gt; \u0026lt;img class=\u0026quot;centerImg\u0026quot; style=\u0026quot; max-width:500px; \u0026quot; src=\u0026quot; /presentations/Alex_Poly/bands/Band.svg\u0026quot;/\u0026gt; Genus of a Seifert surface We have then that as an abstract surface, a Seifert surface for a link is a disc with a number of \u0026ldquo;handles\u0026rdquo;\n($D^1\\times D^1$) added. That number is its genus.\nGenus of a Link We take the smallest genus of possible Seifert surfaces for a link as the genus of the link.\nComputing the genus of a surface \u0026lt;!-- style=\u0026quot; font-size: 150% !important; align-items: center; text-align: center; display: block; margin-left: auto; margin-right: auto;\u0026quot; \u0026ndash;\u0026gt;\n$2g=2-s-n+c$\n$g$: Genus $s$: Number of Seifert circles $n$: Number of components $c$: Number of Crossings \u0026lt;img class=\u0026quot;centerImg\u0026quot; style=\u0026quot; max-width:500px; \u0026quot; src=\u0026quot; /presentations/Alex_Poly/alg/Alg_7.svg\u0026quot;/\u0026gt; Seifert Matrix Link Crossings \u0026lt;img class=\u0026quot;centerImg\u0026quot; style=\u0026quot; max-width:500px; \u0026quot; src=\u0026quot; /presentations/Alex_Poly/crossing/Crossing_\u0026amp;#43;.svg\u0026quot;/\u0026gt; \u0026lt;img class=\u0026quot;centerImg\u0026quot; style=\u0026quot; max-width:500px; \u0026quot; src=\u0026quot; /presentations/Alex_Poly/crossing/Crossing_-.svg\u0026quot;/\u0026gt; Linking number $\\text{Lk}\\LP \\mathscr{L}\\RP=\\text{#} \\img{/presentations/Alex_Poly/crossing/Crossing_+.svg}-\\text{#} \\img{/presentations/Alex_Poly/crossing/Crossing_-.svg}$ Links on a surface We can put oriented simple closed curves through each of the bands.\nA Seifert surface is oriented, it has a top side and bottom side. We can take a push off of each of the curves in the \u0026ldquo;up\u0026rdquo; (blue) direction\nWe can take an $\\LP i,j\\RP$ pair and compute $a_{i,j}=\\text{Lk}\\LP f_i,\\ f_j^+\\RP$ This populates a matrix:\n$$\\begin{bmatrix} \\text{Lk}\\LP f_1,\\ f_1^+\\RP \u0026amp; \\text{Lk}\\LP f_1,\\ f_2^+\\RP \u0026amp; \\cdots \u0026amp; \\text{Lk}\\LP f_1,\\ f_{2g}^+\\RP\\\\ \\text{Lk}\\LP f_2,\\ f_1^+\\RP \u0026amp; \\text{Lk}\\LP f_2,\\ f_2^+\\RP \u0026amp; \\cdots \u0026amp; \\text{Lk}\\LP f_2,\\ f_{2g}^+\\RP\\\\ \\text{Lk}\\LP f_3,\\ f_1^+\\RP \u0026amp; \\text{Lk}\\LP f_3,\\ f_2^+\\RP \u0026amp; \\cdots \u0026amp; \\text{Lk}\\LP f_3,\\ f_{2g}^+\\RP\\\\ \\vdots \u0026amp; \\vdots \u0026amp; \\ddots \u0026amp; \\vdots\\\\ \\text{Lk}\\LP f_{2g},\\ f_1^+\\RP \u0026amp; \\text{Lk}\\LP f_{2g},\\ f_2^+\\RP \u0026amp; \\cdots \u0026amp; \\text{Lk}\\LP f_{2g},\\ f_{2g}^+\\RP\\\\ \\end{bmatrix}$$\nExample \u0026lt;img class=\u0026quot;centerImg\u0026quot; style=\u0026quot; max-width:500px; \u0026quot; src=\u0026quot; /presentations/Alex_Poly/bands/Band.svg\u0026quot;/\u0026gt; \u0026lt;p\u0026gt;$$\\to$$\u0026lt;/p\u0026gt; \u0026lt;!-- A_{rc} --\u0026gt; $$\\begin{bmatrix} -1 \u0026amp; 1 \u0026amp; 0 \u0026amp; 0\\\\ 0 \u0026amp; 1 \u0026amp; 0 \u0026amp; 0\\\\ 0 \u0026amp; 0 \u0026amp; 1 \u0026amp; 1\\\\ 0 \u0026amp; -1 \u0026amp; 0 \u0026amp; 1\\\\ \\end{bmatrix} $$\nAlexander Polynomial For an oriented link $\\mathscr{L}$ and it\u0026rsquo;s associated Seifert matrix $S$ we define the Alexander polynomial of $\\mathscr{L}$ as $\\Delta_\\mathscr{L}\\LP t\\RP=\\text{det}\\LP t^{\\frac{1}{2}}S-t^{-\\frac{1}{2}}S^T\\RP$ $\\operatorname{det}\\LP t^{\\frac{1}{2}}\\begin{bmatrix} -1 \u0026amp; 1 \u0026amp; 0 \u0026amp; 0\\\\ 0 \u0026amp; 1 \u0026amp; 0 \u0026amp; 0\\\\ 0 \u0026amp; 0 \u0026amp; 1 \u0026amp; 1\\\\ 0 \u0026amp; -1 \u0026amp; 0 \u0026amp; 1\\\\ \\end{bmatrix}-t^{\\frac{-1}{2}}\\begin{bmatrix} -1 \u0026amp; 1 \u0026amp; 0 \u0026amp; 0\\\\ 0 \u0026amp; 1 \u0026amp; 0 \u0026amp; 0\\\\ 0 \u0026amp; 0 \u0026amp; 1 \u0026amp; 1\\\\ 0 \u0026amp; -1 \u0026amp; 0 \u0026amp; 1\\\\ \\end{bmatrix}^T\\RP=-t^4+3 t^3-3 t^2+3 t-1 $ $\\begin{bmatrix} -1 \u0026 1 \u0026 0 \u0026 0\\\\ 0 \u0026 1 \u0026 0 \u0026 0\\\\ 0 \u0026 0 \u0026 1 \u0026 1\\\\ 0 \u0026 -1 \u0026 0 \u0026 1\\\\ \\end{bmatrix} $ $$-t^4+3 t^3-3 t^2+3 t-1$$ Invariant $\\Delta_\\mathscr{L}\\LP t\\RP$ is unique up to stabilization, a method for adding bands to the surface. Results in $\\Delta_\\mathscr{L}\\LP t\\RP$ being unique up to a $\\pm t^k$.\nLimitations: Example $$1-t+t^2$$ $$1-t+t^2$$ Bound on genus $\\operatorname{deg}\\LP\\Delta_\\mathscr{L}\\LP t\\RP\\RP\\leq 2\\large g$ Sources Livingston, C. (1993). Knot Theory. Mathematical Association of America. https://doi.org/10.5948/UPO9781614440239 Lickorish, W. B. R. (1997). An Introduction to Knot Theory. In Graduate Texts in Mathematics. Springer New York. https://doi.org/10.1007/978-1-4612-0691-0 Saveliev, N. (2011). Lectures on the Topology of 3-Manifolds. DE GRUYTER. https://doi.org/10.1515/9783110250367 Dale Rolfsen, Knots and links, Mathematics Lecture Series, vol. 7, Publish or Perish, Inc., Houston, TX, 1990, Corrected reprint of the 1976 original. van Wijk, J. J., \u0026 Cohen, A. M. (2006). Visualization of Seifert surfaces. In IEEE Transactions on Visualization and Computer Graphics (Vol. 12, Issue 4, pp. 485-496). Institute of Electrical and Electronics Engineers (IEEE). https://doi.org/10.1109/tvcg.2006.83 van Wijk, J. J., \u0026 Cohen, A. M. (n.d.). Visualization of the Genus of Knots. In VIS 05. IEEE Visualization, 2005. VIS 05. IEEE Visualization, 2005. IEEE. https://doi.org/10.1109/visual.2005.1532843 ","date":"20 February 2023","externalUrl":null,"permalink":"/speaking/misc/alexander_polynomail/","section":"Slides","summary":"Define the Alexander Polynomial","title":"Alexander Polynomial","type":"slides"},{"content":" $\\newcommand{\\N}{\\mathbb{N}} \\newcommand{\\Z}{\\mathbb{Z}} \\newcommand{\\Q}{\\mathbb{Q}} \\newcommand{\\R}{\\mathbb{R}} \\newcommand{\\LP}{\\left(} \\newcommand{\\RP}{\\right)} \\newcommand{\\LS}{\\left\\lbrace} \\newcommand{\\RS}{\\right\\rbrace} \\newcommand{\\LA}{\\left\\langle} \\newcommand{\\RA}{\\right\\rangle} \\newcommand{\\LB}{\\left[} \\newcommand{\\RB}{\\right]} \\newcommand{\\MM}{\\ \\middle|\\ } \\newcommand{\\exp}{\\text{exp}} \\newcommand{\\abs}[1]{\\left\\vert#1\\right\\vert} \\newcommand{\\msr}[1]{m\\left(#1\\right)} \\newcommand{\\inv}[1]{#1^{-1}} \\newcommand{\\bkt}[1]{\\LA \\img{#1}\\RA} \\require{color}$ Summary Below you will find a visualization of a Legendrian unknot in $\\R^3$. You can fly around the space filled with the tangent planes of the standard contact structure. When you approach the unknot you will find the planes tangent to the curve to be colored by the knot.\nThe tooling used to generate this visualization was built as a one off for a topics course, as such I wouldn\u0026rsquo;t call it alpha it\u0026rsquo;s currently a toy proof of concept. That said, the tool is built to support any knot. Unfortunately, right now the knot needs to be defined at compile time (this can be changed in the future if needed/wanted). If you would like another knot or a published (polished FOSS) version of the software let me know.\nInstructions Movement: W - forward A - left S - backward D - right Space - up Shift - down Look around with mouse pointer Note: for some reason the version here spawns the user looking away from the origin. Look around and you\u0026rsquo;ll find the knot. The Visualization ","date":"29 November 2022","externalUrl":null,"permalink":"/resources/general/legendrian_knots/","section":"Resources","summary":"A tool to visualize the Legendrian unknot in the standard contact structure on Rn","title":"Legendrian Unknot Visualization","type":"resources"},{"content":" $\\newcommand{\\N}{\\mathbb{N}} \\newcommand{\\Z}{\\mathbb{Z}} \\newcommand{\\Q}{\\mathbb{Q}} \\newcommand{\\R}{\\mathbb{R}} \\newcommand{\\LP}{\\left(} \\newcommand{\\RP}{\\right)} \\newcommand{\\LS}{\\left\\lbrace} \\newcommand{\\RS}{\\right\\rbrace} \\newcommand{\\LA}{\\left\\langle} \\newcommand{\\RA}{\\right\\rangle} \\newcommand{\\LB}{\\left[} \\newcommand{\\RB}{\\right]} \\newcommand{\\MM}{\\ \\middle|\\ } \\newcommand{\\exp}{\\text{exp}} \\newcommand{\\abs}[1]{\\left\\vert#1\\right\\vert} \\newcommand{\\msr}[1]{m\\left(#1\\right)} \\newcommand{\\inv}[1]{#1^{-1}} \\newcommand{\\bkt}[1]{\\LA \\img{#1}\\RA} \\require{color}$ Discrete Morse Theory Simplicial Complex Simplex A simplex is just an \\(n\\)-dimensional \u0026ldquo;triangle\u0026rdquo; formally\n\\[\\Delta^n=\\LS x \\in \\R^k: x_0+\\cdots+x_{k-1}=1, x_i \\geq 0 \\text { for } i=0, \\ldots, k-1\\RS\\]\nWe can also describe an \\(n\\) simplex combinatorially as a list of the vertices of the simplex \\[\\sigma = \\LB e_0,\\ e_1,\\ \\cdots, e_n\\RB\\]\nFace of a Simplex We define a face \\(\\tau \\), of a simplex \\(\\sigma\\), as a proper subset of \\(\\sigma\\) that is\n\\[ \\tau \u0026lt; \\sigma \\] when \\[ \\LB a,b\\RB \\subset \\LB a, \\ b,\\ c\\RB \\]\nSimplicial complex A simplicial complex is a space \\(K\\) built from glued together simplices and must satisfy\nIf \\(\\tau\\) is the face of a simplex $\\sigma \\in K$ then \\(\\tau\\in K\\) If \\(\\sigma, \\tau \\in K\\) then \\(\\sigma\\cap \\tau\\) is a face of \\(\\sigma\\) and \\(\\tau\\). It\u0026rsquo;s convenient to have a combinatorial description for \\(K\\). This description is called an \u0026ldquo;abstract simplicial complex\u0026rdquo; and is just the collection of combinatorial descriptions of simplices of \\(K\\)\n\\[\\LS \\LB b,c,d\\RB, \\LB b,c\\RB, \\LB b,d\\RB, \\LB c,d\\RB, \\LB a,c\\RB, \\LB a,b\\RB, \\LB d,e\\RB, \\LB a\\RB, \\LB b\\RB, \\LB c\\RB, \\LB d\\RB, \\LB e\\RB\\RS \\]\nDefinition of a \u0026ldquo;Discrete function\u0026rdquo; A discrete function \\(f\\) on a simplicial complex \\(K\\) assigns a number to each simplex\n\\( \\begin{aligned} f\\LP\\LB b,c,d\\RB\\RP \u0026= 5 \\\\ f\\LP\\LB b,c\\RB\\RP \u0026= 4 \\\\ f\\LP\\LB b,d\\RB\\RP \u0026= 5 \\\\ f\\LP\\LB c,d\\RB\\RP \u0026= 3 \\\\ f\\LP\\LB a,c\\RB\\RP \u0026= 1 \\\\ f\\LP\\LB a,b\\RB\\RP \u0026= 1 \\\\ f\\LP\\LB d,e\\RB\\RP \u0026= 5 \\\\ f\\LP\\LB a\\RB\\RP \u0026= 0 \\\\ f\\LP\\LB b\\RB\\RP \u0026= 2 \\\\ f\\LP\\LB c\\RB\\RP \u0026= 2 \\\\ f\\LP\\LB d\\RB\\RP \u0026= 4 \\\\ f\\LP\\LB e\\RB\\RP \u0026= 6 \\\\ \\end{aligned} \\) Definition of a \u0026ldquo;Discrete Morse function\u0026rdquo; A discrete function \\[f:K\\to \\R\\] is called a discrete Morse function if for every \\(n\\)-simplex \\(\\alpha^n \\in K\\) the following are true\n\\(\\#\\LS \\alpha^n\u0026lt;\\beta^{n+1} \\MM f\\LP\\beta^{n+1}\\RP\\leq f\\LP\\alpha^n\\RP\\RS\\leq 1\\) \\(\\#\\LS \\alpha^n\u0026gt;\\gamma^{n-1} \\MM f\\LP\\gamma^{n-1}\\RP\\geq f\\LP\\alpha^n\\RP\\RS\\leq 1\\) \\(\\#\\LS \\alpha^n\u003c\\beta^{n+1} \\MM f\\LP\\beta^{n+1}\\RP\\leq f\\LP\\alpha^n\\RP\\RS\\leq 1\\) \\(\\#\\LS \\alpha^n\u003e\\gamma^{n-1} \\MM f\\LP\\gamma^{n-1}\\RP\\geq f\\LP\\alpha^n\\RP\\RS\\leq 1\\) \\(\\#\\LS \\alpha^n\u003c\\beta^{n+1} \\MM f\\LP\\beta^{n+1}\\RP\\leq f\\LP\\alpha^n\\RP\\RS\\leq 1\\) \\(\\#\\LS \\alpha^n\u003e\\gamma^{n-1} \\MM f\\LP\\gamma^{n-1}\\RP\\geq f\\LP\\alpha^n\\RP\\RS\\leq 1\\) Definition of a Critical simplex An \\(n-\\)simplex \\(\\alpha^n\\), of a complex \\(K\\), is critical under a discrete Morse function \\(f\\) if the following are true\n\\(\\#\\LS \\alpha^n\u0026lt;\\beta^{n+1} \\MM f\\LP\\beta^{n+1}\\RP\\leq f\\LP\\alpha^n\\RP\\RS=0\\) \\(\\#\\LS \\alpha^n\u0026gt;\\gamma^{n-1} \\MM f\\LP\\gamma^{n-1}\\RP\\geq f\\LP\\alpha^n\\RP\\RS=0\\) We should note that for all the \\(\\alpha\\in K\\) at least one of \\(1.\\) and \\(2.\\) must be true.\n\\(\\#\\LS \\alpha^n\u003c\\beta^{n+1} \\MM f\\LP\\beta^{n+1}\\RP\\leq f\\LP\\alpha^n\\RP\\RS=0\\) \\(\\#\\LS \\alpha^n\u003e\\gamma^{n-1} \\MM f\\LP\\gamma^{n-1}\\RP\\geq f\\LP\\alpha^n\\RP\\RS=0\\) Discrete Gradient Vector Field A discrete gradient vector field on a simplicial complex \\(K\\) is a collection of pairs \\(V=\\LS\\tau^n\u0026lt;\\sigma^{n+1}\\RS\\) of simplices in \\(K\\) such that each simplex is in at most one pair of \\(V\\).\nA discrete Morse function induces a gradient vector field on \\(K\\).\nIf \\(\\alpha\\) is a non-critical simplex, and \\[\\alpha\u0026lt;\\beta\\] with \\[f\\LP\\beta\\RP\u0026lt;f\\LP\\alpha\\RP\\] We draw an arrow from \\(\\alpha\\to \\beta\\).\nSimplicial collapse The simplicial equivalent of a deformation retraction is called a simplicial collapse.\nMaximal Face A face \\(\\sigma\\in K\\) is called maximal if there is no \\(\\beta\\in K\\) so that \\(\\sigma\u003c\\beta\\). Free face A simplex \\(\\tau\\) of a simplicial complex \\(K\\) is called a free face if \\(\\tau\\) is a face of exactly one maximal face. Collapse If \\(\\tau\\) is a free face of \\(\\sigma\\) we can collapse \\(K \\searrow K^\\prime\\) by the following set operation\n\\[K^\\prime = K-\\LP\\tau \\cup \\sigma\\RP\\]\nCollapsing A Vector Field Given a vector field we can simplicially collapse following the arrows.\nMain result of discrete Morse Theory Suppose \\(K\\) is a simplicial complex with a discrete Morse function. Then \\(K\\) is homotopy equivalent to a CW complex with exactly one cell of dimension \\(n\\) for each critical simplex of dimension \\(n\\).\nSphere Theorems Star of a face The star of a face \\(\\tau\\) of \\(K\\) is the subcomplex of \\(K\\) consisting of all faces \\(\\sigma\\) of \\(K\\) where \\(\\tau \u0026lt; \\sigma\\), as well as all faces of \\(\\sigma\\).\nLink of a face The link of a face \\(\\tau\\) of \\(K\\) is the subcomplex of \\(K\\) consisting of all faces of the star of \\(\\tau\\) that do not intersect \\(\\tau\\).\nCombinatorial d-ball A complex \\(K\\) is a combinatorial d-ball if \\(K\\) and \\(\\Delta^d\\) have isomorphic subdivisions.\nA combinatorial \\(\\LP d-1\\RP\\)-sphere is the boundary of a \\(d\\)-ball\ncombinatorial d-manifold A complex is a \\(d\\)-manifold if the link of every vertex is either a \\((d-1)\\)-ball or \\((d-1)\\)-sphere.\nSphere theorems Let \\(K\\) be a combinitorial \\(d\\)-manifold with boundary which simplicially collapses to a vertex. Then \\(K\\) is a combinitorial \\(d\\)-ball.\nLet \\(X\\) be a combinatorial \\(d\\)-manifold with a discrete with a discrete Morse function with exactly two critical simplices. Then \\(X\\) is a combinitorial sphere.\nReferences Robin Forman, A user\u0026rsquo;s guide to discrete Morse theory Robin Forman, Morse Theory for Cell Complexes Jean Gallier \u0026amp; Jocelyn Quaintance, Aspects of Convex Geometry ","date":"4 October 2022","externalUrl":null,"permalink":"/speaking/misc/discrete_morse_theory/","section":"Slides","summary":"Brief overview of Discrete Morse Theory","title":"UIowa Topology seminar - Discrete Morse Theory","type":"slides"},{"content":"","externalUrl":null,"permalink":"/cv/joe-starr/","section":"Cvs","summary":"","title":"","type":"cv"},{"content":"","externalUrl":null,"permalink":"/resume/joe-starr/","section":"Resumes","summary":"","title":"","type":"resume"},{"content":"","externalUrl":null,"permalink":"/authors/","section":"Authors","summary":"","title":"Authors","type":"authors"},{"content":"","externalUrl":null,"permalink":"/categories/","section":"Categories","summary":"","title":"Categories","type":"categories"},{"content":"","externalUrl":null,"permalink":"/cv/","section":"Cvs","summary":"","title":"Cvs","type":"cv"},{"content":"","externalUrl":null,"permalink":"/resume/","section":"Resumes","summary":"","title":"Resumes","type":"resume"}]