Metabelian SL(n,C) representations of knot groups II: fixed points

TL;DR

This paper characterizes fixed points of a cyclic group action on SL(n,C) character varieties of knot groups, linking them to metabelian representations and showing the associated twisted Alexander polynomial is a polynomial in t^n.

Contribution

It identifies fixed points of the cyclic action in terms of metabelian characters and proves the twisted Alexander polynomial is a polynomial in t^n for these representations.

Findings

Fixed points correspond to characters of metabelian representations.

Twisted Alexander polynomial for these representations is a polynomial in t^n.

Provides a new understanding of the structure of character varieties under group actions.

Abstract

Given a knot K in an integral homology sphere with exterior N_K, there is a natural action of the cyclic group Z/n on the space of SL(n,C) representations of the knot group \pi_1(N_K), and this induces an action on the SL(n,C) character variety. We identify the fixed points of this action in terms of characters of metabelian representations, and we apply this to show that the twisted Alexander polynomial associated to an irreducible metabelian SL(n,C) representation is actually a polynomial in t^n.