Asymptotic behavior of twisted Alexander invariants for hyperbolic knots with at most six crossings

Abstract

Let $K$ be a hyperbolic knot and let $ρ_n$ be the $n$-dimensional irreducible representation induced from a lift of its holonomy representation. Motivated by Goda's asymptotic volume formula and the complexified Volume Conjecture, we study whether the higher-dimensional twisted Alexander invariants associated with $ρ_n$ detect the complex volume of the knot complement. We compute $$ \fracπ{2} \log \left( \frac{A_{K,n-2}(1)A_{K,n+2}(1)} {A_{K,n}(1)^2} \right) $$ for all hyperbolic knots with at most six crossings. Our numerical experiments indicate that these values approach $$ \operatorname{Vol}(S^3\setminus K) +i\,2π^2\operatorname{CS}(S^3\setminus K) $$ modulo $iπ^2\mathbb{Z}$. Based on these computations, we propose a complexified analogue of Goda's asymptotic volume formula.