Fibering of double twist knots via the adjoint hyperbolic torsion polynomial

TL;DR

This paper investigates the adjoint hyperbolic torsion polynomial for double twist knots, demonstrating that it encodes key topological features such as genus and fibering within this family of rational knots.

Contribution

It introduces the use of the adjoint hyperbolic torsion polynomial to determine genus and fibering for double twist knots, a novel application in knot theory.

Findings

The polynomial determines the genus of double twist knots.

The polynomial detects whether a double twist knot is fibered.

Application of algebraic integers in analyzing knot properties.

Abstract

For a hyperbolic knot $K$ in $S^3$, the adjoint hyperbolic torsion polynomial $\mathcal T^{\mathrm{Ad}}_K(t) \in \mathbb C[t^{\pm 1}]$ is defined as a normalization of the twisted Alexander polynomial of $K$ associated with the $\mathrm{SL}_3(\mathbb C)$-representation obtained by composing the holonomy representation of $K$ with the adjoint action of $\mathrm{SL}_2(\mathbb C)$ on its Lie algebra $\mathfrak{sl}_2(\mathbb C)$. In this paper we consider the adjoint hyperbolic torsion polynomial for a two-parameter family of rational knots called double twist knots, and show that $\mathcal T^{\mathrm{Ad}}_K(t)$ determines the genus and fibering of this family by using algebraic integers.