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

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