Asymptotics of twisted Alexander polynomials and hyperbolic volume

Understand

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.