Representations of knot groups into $\mathrm{SL}_n(\mathbf{C})$ and twisted Alexander polynomials

TL;DR

This paper investigates the deformation theory of knot group representations into SL_n(C), linking the existence of irreducible deformations to twisted Alexander polynomials and analyzing the local structure of associated varieties.

Contribution

It provides necessary and sufficient conditions, based on twisted Alexander polynomials, for the existence of irreducible deformations of certain reducible representations of knot groups.

Findings

Necessary conditions for irreducible deformations are established.

Sufficient conditions for irreducible deformations are provided.

A duality theorem for twisted Alexander polynomials is proved.

Abstract

Let $\Gamma$ be the fundamental group of the exterior of a knot in the three-sphere. We study deformations of representations of $\Gamma$ into $\mathrm{SL}_n(\mathbf{C})$ which are the sum of two irreducible representations. For such representations we give a necessary condition, in terms of the twisted Alexander polynomial, for the existence of irreducible deformations. We also give a more restrictive sufficient condition for the existence of irreducible deformations. We also prove a duality theorem for twisted Alexander polynomials and we describe the local structure of the representation and character varieties.