Twisted Alexander polynomials and a partial order on the set of prime knots

TL;DR

This paper surveys recent research connecting twisted Alexander polynomials with a partial order on prime knots, providing examples and open problems in the field.

Contribution

It introduces a new perspective on prime knots using twisted Alexander polynomials and explores their partial ordering, highlighting recent developments and open questions.

Findings

Examples illustrating the partial order on prime knots

Identification of open problems in the area

Connections between twisted Alexander polynomials and knot classification

Abstract

We give a survey of some recent papers by the authors and Masaaki Wada relating the twisted Alexander polynomial with a partial order on the set of prime knots. We also give examples and pose open problems.