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Use Case

Functionality

Mathematical Description

The rational tangle computation module computes 3 pieces of data for a rational tangles; rational number, parity, and algebraic equivalence of closures.

Rational Number

In Conway's original tangle paper 1 he states that rational tangles and rational numbers are in one to one correspondence, this was later proven for tangles by Goldman and Kauffman 2. The correspondence comes from interpreting the twist vector of a rational tangle as a finite continued fraction that is: \(\LB a\ b\ c\RB=c+\frac{1}{b+\frac{1}{a}}\)

Parity

The parity of a rational tangle describes it's fixed point connectivity

Example

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the connectivity, formally described by Kauffman and Lambropoulou 3 as

Kauffman, Lambropoulou

A rational tangle \(T\) has connectivity type \(\asymp\) if and only if its fraction has parity e/o. \(T\) has connectivity type \(><\) if and only if its fraction has parity o/e. \(T\) has connectivity type \(\chi\) if and only if its fraction has parity o/o. (Note that the formal fraction of \([\infty]\) itself is \(1 / 0\).) Thus the link \(N(T)\) has two components if and only if \(T\) has fraction \(F(T)\) of parity e/o.

Algebraic Equivalence

The algebraic equivalence class for a tangle describes the knot equivalence class of the numerator closure of the tangle. Equivalence is give by Schubert 4 as:

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 \(0<p\). 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. \(q \equiv q^{\prime}(\bmod p)\) or \(q q^{\prime} \equiv 1(\bmod p)\).

where \(N()\) indicates the numerator closure of the object tangle.

  1. J.H. Conway. An enumeration of knots and links, and some of their algebraic properties. Computational Problems in Abstract Algebra, pages 329–358, 1970. URL: https://linkinghub.elsevier.com/retrieve/pii/B9780080129754500345 (visited on 2023-05-04), doi:10.1016/B978-0-08-012975-4.50034-5. ↩

  2. Jay R. Goldman and Louis H. Kauffman. Rational Tangles. Advances in Applied Mathematics, 18(3):300–332, 1997. URL: https://linkinghub.elsevier.com/retrieve/pii/S0196885896905114 (visited on 2024-08-16), doi:10.1006/aama.1996.0511. ↩

  3. Louis H. Kauffman and Sofia Lambropoulou. On the Classification of Rational Knots. arXiv: Geometric Topology, 2002. URL: https://arxiv.org/abs/math/0212011 (visited on 2023-05-26), doi:10.48550/ARXIV.MATH/0212011. ↩

  4. Horst Schubert. Knoten mit zwei brücken. Mathematische Zeitschrift, 65:133–170, 1956. URL: http://eudml.org/doc/169591. ↩