Let G be a complex reductive algebraic group. In this talk, we will
discuss a general conjecture that there is a natural action of the
double affine Hecke algebra of type G on a (quantized) character
variety of the complement of a knot in S^3. We will explain motivation
and give an explicit construction in the case of SL(2,C). In that case,
we will show that the classical limit (q=1) of our conjecture follows from (and essentially reduces to) a known conjecture about the structure of knot groups due to Brumfiel and Hilden (1990). Time permitting, we
will also discuss some examples and implications of our conjecture.
The main implication is the existence of 3-variable polynomial knot
invariants that specialize to the famous Jones polynomial and its
colored versions introduced by Witten, Reshetikhin and Turaev.
(The talk is based on joint work with P. Samuelson)