Fibering of double twist knots via the adjoint hyperbolic torsion polynomial

Source abstract

Abstract For a hyperbolic knot K $K$ upper K in S 3 $S^3$ upper S cubed , the adjoint hyperbolic torsion polynomial T K Ad ( t ) ∈ C [ t ± 1 ] $\mathcal T^{\mathrm {Ad}}_K(t) \in \mathbb C[t^{\pm 1}]$ script upper T Subscript upper K Superscript upper A d Baseline left parenthesis t right parenthesis element of double struck upper C left bracket t Superscript plus or minus 1 Baseline right bracket is defined as a normalization of the twisted Alexander polynomial of K $K$ upper K associated with the SL 3 ( C ) $\mathrm {SL}_3(\mathbb C)$ upper S upper L 3 left parenthesis double struck upper C right parenthesis -representation obtained by composing the holonomy representation of K $K$ upper K with the adjoint action of SL 2 ( C ) $\mathrm {SL}_2(\mathbb C)$ upper S upper L 2 left parenthesis double struck upper C right parenthesis on its Lie algebra sl 2 ( C ) $\mathfrak {sl}_2(\mathbb C)$ German sl 2 left parenthesis double struck upper C right parenthesis . In this article, we consider the adjoint hyperbolic torsion polynomial for a two-parameter family of rational knots called double twist knots and show that T K Ad ( t ) $\mathcal T^{\mathrm {Ad}}_K(t)$ script upper T Subscript upper K Superscript upper A d Baseline left parenthesis t right parenthesis determines the genus and fibering of this family by using algebraic integers.