Twisted Alexander vanishing groups of knots

TL;DR

This paper explores the properties of twisted Alexander vanishing (TAV) groups in knot theory, analyzing their orders, constructing knots with vanishing polynomials, and examining the impact of faithful irreducible representations.

Contribution

It introduces new results on the orders of TAV groups, constructs specific knots with vanishing polynomials, and links faithful irreducible representations to polynomial vanishing.

Findings

Determined the orders of TAV groups.

Constructed knots with vanishing twisted Alexander polynomials.

Showed that faithful irreducible representations lead to polynomial vanishing.

Abstract

In our previous work, we introduced the notion of a twisted Alexander vanishing (TAV) group, defined as a finite group for which the corresponding twisted Alexander polynomial of a knot vanishes. In this paper, we discuss the orders of TAV groups and construct knots whose twisted Alexander polynomials vanish. Moreover, we show that every faithful irreducible representation of a TAV group causes the twisted Alexander polynomial to be zero.