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1: Rational Tangles

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Learning Objectives
    • Discover how different types of twists on a tangle determine its tangle number.
    • Argue for why the arithmetic of the rational numbers makes certain relationships among tangle numbers necessary.
    • Calculate the fraction of a rational tangle in two different ways, and argue for why the fraction is an invariant of rational tangles.

    "What I like doing is taking something that other people thought was complicated and difficult to understand, and finding a simple idea. So that any fool - and, in this case, you - can understand the complicated thing. ---John Conway"

    As we discovered in our first class, crossings are one of the first ways for us to understand the connections between knots and algebra: somehow, if we can say "enough" about how a strand crosses itself, we can characterize the essential nature of a knot. So we'll begin by focusing as much as possible only on crossings, by studying objects known as tangles, in which crossings are created between two strands by twisting up their endpoints.


    Thumbnail: A knot diagram of the trefoil knot, the simplest non-trivial knot. (Public Domain; Marnanel via Wikipedia)

    This page titled 1: Rational Tangles is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Matthew Salomone via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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