On the number of components of twisted torus links

TL;DR

This paper introduces an algorithm inspired by the Euclidean algorithm to determine the number of components in twisted torus links, generalizing the known gcd-based component count for torus links.

Contribution

It provides a novel computational method for the component count of twisted torus links and establishes a gcd-based criterion for when these links are knots.

Findings

Number of components is a multiple of gcd(p, q, r, s).

T(p, q; r, s) is a knot iff gcd(p, q, r, s) = 1.

Proved several conjectures on component counts in twisted torus 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.