The twisted Alexander polynomial for finite abelian covers over three manifolds with boundary

TL;DR

This paper generalizes the computation of Alexander polynomials to finite abelian covers of 3-manifolds with boundary, extending known formulas from knot theory to broader classes of manifolds.

Contribution

It introduces a formula for twisted Alexander polynomials of finite abelian covers over 3-manifolds with boundary, broadening the scope of classical knot invariants.

Findings

Provides explicit formulas for twisted Alexander polynomials in this setting

Extends classical results from cyclic covers to abelian covers

Enhances understanding of 3-manifold invariants with boundary

Abstract

We provide the twisted Alexander polynomials of finite abelian covers over three-dimensional manifolds whose boundary is a finite union of tori. This is a generalization of a well-known formula for the usual Alexander polynomial of knots in finite cyclic branched covers over the three-dimensional sphere.