Deformations of the peripherial map for knot complements

Peter Samuelson (Toronto)

Abstract

Deformations of the peripherial map for knot complements Abstract: The space $$Rep(M)$$ of representations of the fundamental group $$\pi_1(M)$$ of a 3-manifold M into $$SL_2(\mathbb{C})$$ has played an important role in the study of 3-manifolds. If $$M = S^3 \setminus K$$ is the complement of a knot in the 3-sphere, then there is a map $$Rep(M) \to Rep(T^2)$$ given by restricting representations to the boundary. There is a natural deformation $$X(s,t)$$ of the space $$Rep(T^2)$$ depending on two complex parameters which comes from a "double affine Hecke algebra." We will discuss some background and then describe a conjecture that the map $$Rep(M) \to Rep(T^2)$$ has a canonical deformation to a map $$Rep(M) \to X(s,t)$$. (This is joint work with Yuri Berest.)