Fibering of double twist knots via the adjoint hyperbolic torsion polynomial

Abstract

For a hyperbolic knot $K$ in $S^3$, the adjoint hyperbolic torsion polynomial ${\mathcal{T}}_K^{\mathrm{Ad}}(t)\in \mathbb{C}[t^{\pm 1}]$ 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 ${\mathcal{T}}_K^{\mathrm{Ad}}(t)$ determines the genus and fibering of this family by using algebraic integers.

14 pages