The Jones polynomial for a torus knot with twists

TL;DR

This paper computes the Jones polynomial for a specific family of twisted torus links and knots, establishing conditions under which the polynomial is trivial, thereby providing insights into their topological properties.

Contribution

It introduces a method to compute the Jones polynomial for twisted torus links and characterizes when these polynomials are trivial, extending understanding of their knot invariants.

Findings

Jones polynomial computed explicitly for the family $T(p,q,2,n)$

Trivial Jones polynomial implies the knot is trivial in this family

Provides a criterion linking polynomial triviality to knot triviality

Abstract

We compute the Jones polynomial for a three-parameter family of links, the twisted torus links of the form $T((p,q),(2,s))$ where $p$ and $q$ are coprime and $s$ is nonzero. When $s = 2n$, these links are the twisted torus knots $T(p,q,2,n)$. We show that for $T(p,q,2,n)$, the Jones polynomial is trivial if and only if the knot is trivial.