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

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