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 $\mathcal T^{\mathrm{Ad}}_K(t) \in \mathbb C[t^{\pm 1}]$ 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.