Representations of knot groups into $\mathrm{SL}_n(\mathbf{C})$ and twisted Alexander polynomials
Résumé
Let $\Gamma$ be the fundamental group of the exterior of a knot in the three-sphere. For a representation of $\Gamma$ in $\mathrm{SL}_n(\mathbf{C})$ that is direct sum of two irreducible ones, we give necessary and sufficient conditions for the existence of deformations to irreducible representations, in terms of twisted Alexander polynomials. We also prove a duality theorem for twisted Alexander polynomials and we describe the local structure of the representation and character varieties.