The Alexander polynomial of twisted torus knots

TL;DR

This paper derives an explicit formula for the Alexander polynomial of twisted torus knots using knot group presentations and Fox calculus, providing insights into their genus bounds and L-space knot properties.

Contribution

It introduces a new explicit formula for the Alexander polynomial of twisted torus knots, expanding understanding of their algebraic and topological properties.

Findings

Provides a lower bound for the genus of certain twisted torus knots

Identifies families of twisted torus knots that are not L-space knots

Uses Fox calculus to derive the Alexander polynomial formula

Abstract

Twisted torus knots are a generalization of torus knots, obtained by introducing additional full twists to adjacent strands of the torus knots. In this article, we present an explicit formula for the Alexander polynomial of twisted torus knots. Our approach utilizes a presentation of the knot group of twisted torus knots combined with Fox's free differential calculus. As applications, we provide a lower bound for the genus of certain families of twisted torus knots and identify families of twisted torus knots that are not $L$-space knots.