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On the number of components of twisted torus links

Adnan, Thiago de Paiva, Kyungbae Park

TL;DR

The paper addresses the problem of determining the number of components of twisted torus links $T(p,q;r,s)$, which is not immediately determined by the parameters. It introduces a Euclidean-algorithm-like procedure based on a sequence of quadruples $(p_i,q_i,r_i,s_i)$ and component-preserving moves that reduce the calculation to a final step. Key results include that $NC(p,q;r,s)$ is a positive multiple of $ vert ext{gcd}(p,q,r,s) vert$ and that $T(p,q;r,s)$ is a knot only when $ ext{gcd}(p,q,r,s)=1$, with the algorithm terminating in a terminal case (either $q_n=0$ or $s_n=0$) to yield $NC(p_n,q_n,r_n,s_n)$; moreover, explicit checks and conjecture verifications from LC-2016 are obtained for several families. The method further generalizes to $T$-links with three parameter pairs, giving an algorithm to compute $NC$ in that setting, while not yielding a known closed-form for all cases and suggesting open questions for higher-parameter links.

Abstract

Twisted torus links $T(p,q;r,s)$ generalize torus links by introducing $s$ additional twists on $r$ adjacent strands of the torus link $T(p,q)$. It is well known that the number of components of a torus link $T(p, q)$ is given by the greatest common divisor of $p$ and $q$. However, determining the number of components of twisted torus links is not as straightforward based solely on their parameters. In this work, we present a Euclidean algorithm-like procedure for computing the number of components of twisted torus links based on their parameters. As a result, we show that the number of components of a twisted torus link $T(p, q; r, s)$ is a multiple of $\gcd(p, q, r, s)$, and in particular, $T(p, q; r, s)$ is a knot only if $\gcd(p, q, r, s) = 1$. We also use our algorithm to prove several conjectures related to the number of components in twisted torus links.

On the number of components of twisted torus links

TL;DR

The paper addresses the problem of determining the number of components of twisted torus links , which is not immediately determined by the parameters. It introduces a Euclidean-algorithm-like procedure based on a sequence of quadruples and component-preserving moves that reduce the calculation to a final step. Key results include that is a positive multiple of and that is a knot only when , with the algorithm terminating in a terminal case (either or ) to yield ; moreover, explicit checks and conjecture verifications from LC-2016 are obtained for several families. The method further generalizes to -links with three parameter pairs, giving an algorithm to compute in that setting, while not yielding a known closed-form for all cases and suggesting open questions for higher-parameter links.

Abstract

Twisted torus links generalize torus links by introducing additional twists on adjacent strands of the torus link . It is well known that the number of components of a torus link is given by the greatest common divisor of and . However, determining the number of components of twisted torus links is not as straightforward based solely on their parameters. In this work, we present a Euclidean algorithm-like procedure for computing the number of components of twisted torus links based on their parameters. As a result, we show that the number of components of a twisted torus link is a multiple of , and in particular, is a knot only if . We also use our algorithm to prove several conjectures related to the number of components in twisted torus links.
Paper Structure (7 sections, 21 theorems, 38 equations, 5 figures)

This paper contains 7 sections, 21 theorems, 38 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Moves preserving the number of components
  3. Examples
  4. Case: $r=2$
  5. Case: $\gcd(p,q)\geq r$
  6. Case: $s=\pm q$
  7. $T$-links with three pairs of parameters

Key Result

Theorem 1.1

For positive integers $p\geq r> 0$ and $q,s\in\mathbb{Z}$, let $NC(p,q;r,s)$ denote the number of components of the twisted torus link $T(p,q;r,s)$, and $[x]_m$ denote the residue of $x$ modulo $m$. Define and let $\{(p_i, q_i, r_i, s_i)\}_{i=1}^n$ be the sequence of quadruples obtained by the following recursive procedure: If $q_{i} = 0$ or $s_{i} = 0$, let $n = i$ . Otherwise, Then $\{(p_i, q_

Figures (5)

  • Figure 1: A diagram of the twisted torus link $T(9,6;7,4)$.
  • Figure 2: An isotopy that horizontally shifts the additional twist region.
  • Figure 3: A transformation of $T(6,4;3,2)$ to $T(4,6;3,-2)$ on a thickened flat torus.
  • Figure 4: A transformation from $T(8,4;5,3)$ to $T(5,7;4,-3)$ that preserves the number of components.
  • Figure 5: A transformation from $T(8,4;6,3;5,3)$ to $T(4,-2;6,7;5,3)$ that preserves the number of components.

Theorems & Definitions (38)

  • Theorem 1.1
  • Remark
  • Theorem 1.2
  • proof
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 28 more