Twisted Alexander vanishing groups of knots

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.