Asymptotics of twisted Alexander polynomials and hyperbolic volume

TL;DR

This paper investigates the asymptotic behavior of twisted Alexander polynomials for hyperbolic knots and manifolds, revealing their connection to hyperbolic volume and Mahler measures, and extends known results to more general cases.

Contribution

It establishes the asymptotic behavior of twisted Alexander polynomials at roots of unity for hyperbolic knots and manifolds, linking them to hyperbolic volume and Mahler measures, with a unified proof approach.

Findings

Asymptotic formulas relate twisted Alexander polynomials to hyperbolic volume.

Extension of asymptotic behavior to cusped hyperbolic manifolds.

Deduction of asymptotics for Mahler measures of these polynomials.

Abstract

For a hyperbolic knot and a natural number n, we consider the Alexander polynomial twisted by the n-th symmetric power of a lift of the holonomy. We establish the asymptotic behavior of these twisted Alexander polynomials evaluated at unit complex numbers, yielding the volume of the knot exterior. More generally, we prove the asymptotic behavior for cusped hyperbolic manifolds of finite volume. The proof relies on results of M\"uller, and Menal-Ferrer and the last author. Using the uniformity of the convergence, we also deduce a similar asymptotic result for the Mahler measures of those polynomials.