Hyperbolic Volume and Twisted Alexander invariants of Knots and Links

TL;DR

This paper investigates the relationship between hyperbolic volume and twisted Alexander invariants of knots and links, revealing how the asymptotic behavior of these invariants encodes geometric information about the link complement.

Contribution

It introduces a method to extract hyperbolic volume from the asymptotics of twisted Alexander polynomials associated with certain representations.

Findings

Hyperbolic volume can be recovered from the asymptotic behavior of twisted Alexander invariants.

The sequence of invariants evaluated at specific points encodes geometric information.

The approach links algebraic invariants to hyperbolic geometry.

Abstract

Let $\Delta_{L,\rho_n}(t)$ be the twisted Alexander polynomial with respect to the representation given by the composition of the lift of the holonomy representation of a certain hyperbolic link $L$ and the $n$-dimensional irreducible complex representation of $\text{SL}(2,\mathbb C)$. We consider a sequence of $\Delta_{L,\rho_n}(t)$ and extract the volume of the complement of $L$ from the asymptotic behaviour of the sequence obtained by evaluating $t=1$ or $t=-1$.