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

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.

References in corpus (1)

• Deformations of reducible representations of knot groups into $\mathrm{SL}(n,\mathbf{C})$

Cited by in corpus (5)

• Geometry of the SL(3,C)-character variety of torus knots
• Discrete isometry groups of symmetric spaces
• Dimension of representation and character varieties for two and three-orbifolds
• Torsion function on character varieties
• Reidemeister torsion form on character varieties