A survey of twisted Alexander polynomials

TL;DR

This paper surveys the theory of twisted Alexander polynomials for 3-manifolds, discussing their properties, relation to Reidemeister torsion, and applications in topology, including higher order variants.

Contribution

It provides a comprehensive overview of twisted Alexander polynomials, highlighting their formal properties, relationships, and diverse applications in topological research.

Findings

Summarizes formal properties of twisted Alexander polynomials

Explains their relationship to twisted Reidemeister torsion

Reviews applications to topological problems

Abstract

We give a short introduction to the theory of twisted Alexander polynomials of a 3--manifold associated to a representation of its fundamental group. We summarize their formal properties and we explain their relationship to twisted Reidemeister torsion. We then give a survey of the many applications of twisted invariants to the study of topological problems. We conclude with a short summary of the theory of higher order Alexander polynomials.