  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.
