SL(2,C)-character varieties of knots and maps of degree 1

We ask to what extend the SL(2,C)-character variety of the fundamental group of the complement of a knot in S^3 determines the knot. Our methods use results from group theory, classical 3-manifold topology, but also geometric input in two ways: the geometrisation theorem for 3-manifolds, and instanton gauge theory. In particular this is connected to SU(2)-character varieties of two-component links, a topic where much less is known than in the case of knots. This is joint work with Michel Boileau, Teruaki Kitano, and Steven Sivek.