Twisted Alexander Invariants of Twisted Links

TL;DR

This paper investigates how twisted Alexander invariants of links in 3-spheres change under certain surgeries, providing new insights into their behavior for finite-image representations as the surgery parameter grows.

Contribution

It introduces new results on the behavior of twisted Alexander polynomials of links after 1/q-surgery, focusing on finite-image representations and their asymptotic properties.

Findings

Invariants are explicitly computed for links after surgery.

Behavior of invariants as q approaches infinity is characterized.

Results connect link invariants with topological changes due to surgery.

Abstract

Let L be an oriented (d+1)-component link in the 3-sphere, and let L(q) be the d-component link in a homology 3-sphere that results from performing 1/q-surgery on the last component. Results about the Alexander polynomial and twisted Alexander polynomials of L(q) corresponding to finite-image representations are obtained. The behavior of the invariants as q increases without bound is described.