Twisted Alexander polynomials of a knot for group extensions

TL;DR

This paper develops formulas for twisted Alexander polynomials of knots related to group extensions, aiding the study of knot properties like fiberedness and vanishing groups.

Contribution

It introduces mod p formulas and considers central extensions to analyze twisted Alexander polynomials for knots in new contexts.

Findings

Derived a mod p formula for twisted Alexander polynomials using the regular representation.

Analyzed twisted Alexander polynomials for central extensions of finite groups.

Applied formulas to study vanishing groups and orders for non-fibered knots.

Abstract

In this paper, we discuss twisted Alexander polynomials of a knot for group extensions of a finite group in two directions. Firstly, we provide a mod $p$ formula for the twisted Alexander polynomial of a knot in the $3$-sphere associated with the regular representation of a finite group. Secondly, we consider twisted Alexander polynomials of a knot for a series of central extensions of a finite group. Moreover, we apply these formulas for twisted Alexander polynomials to the study of twisted Alexander vanishing groups and orders for non-fibered knots.