Fibering of double twist knots via the adjoint hyperbolic torsion polynomial

2025/08/20 by Anh T. Tran, Tran, Anh T.
Mathematics · Computer Science · #Geometric and Algebraic Topology #semigroups and automata theory #Advanced Combinatorial Mathematics

paper · pdf · doi:10.48550/arxiv.2508.14731

Abstract

For a hyperbolic knot K in S³, the adjoint hyperbolic torsion polynomial \mathcal T^Ad_K(t) ∈ \mathbb C[t^{± 1}] is defined as a normalization of the twisted Alexander polynomial of K associated with the SL₃(\mathbb C)-representation obtained by composing the holonomy representation of K with the adjoint action of SL₂(\mathbb C) on its Lie algebra \mathfraksl₂(\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^Ad_K(t) determines the genus and fibering of this family by using algebraic integers.

Citations

• On a conjecture of Dunfield, Friedl and Jackson
• Thurston norm, fibered manifolds and twisted Alexander polynomials
• Reidemeister torsion, twisted Alexander polynomial and fibered knots

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