2011-04-29

Deformations of Representation Varieties of Knot Groups

Peter Samuelson, Cornell University

If \pi is a finitely generated discrete group and G is an algebraic group, the set Hom(\pi, G) has a natural structure of an algebraic variety that comes with an algebraic action of G (by conjugation). I will discuss a topological construction of the ring of invariant functions on Hom(\pi, G) for G = SL_2(C). If \pi is the fundamental group of a 3-manifold M, this construction admits a noncommutative deformation K_q(M) motivated by knot theory. If F is the boundary of M, then K_q(F x [0,1]) deforms as an algebra and K_q(M) becomes a left module over K_q(F x [0,1]). In the case when M = S^3\K is a knot complement, the module K_q(M) generalizes a classical knot invariant - the so-called A-polynomial introduced by Cooper, Culler, Gillett, Long and Shalen. In this talk, I will explain a generalization of the topological construction of K_q(M) that produces for each knot K a module over the double affine Hecke algebra (DAHA) introduced by Cherednik.