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

Michael Heusener and Joan Porti

Vol. 277 (2015), No. 2, 313–354

DOI: 10.2140/pjm.2015.277.313

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 SL${}_n\left(C\right)$ 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.

Keywords

variety of representations, character variety, twisted Alexander polynomial, deformations

Mathematical Subject Classification 2010

Primary: 57M25, 57M05

Secondary: 57M27

Milestones

Received: 14 July 2014

Revised: 11 March 2015

Accepted: 12 March 2015

Published: 15 September 2015