The deformation variety of a knot is the complex curve of SL_2(C) representations of the knot complement, viewed from the boundary. The deformation variaty of a knot is an affine curve in C^2 which is defined by the zeros of a 2-variable polynomial with integer coefficients; the so-called A-polynomial of a knot.
The characteristic variety of a knot is a new object determined entirely from the Jones polynomial of a knot and its parallels. The deformation variety of a knot is also a complex curve in C^2, defined by the so-called specialization of the non-commutative A-polynomial, when q=1.
Last spring we conjectured that the deformation and characteristic varieties of a knot are equal, and proved it in the case of the trefoil and figure 8 knots (the former is not hyperbolic, the latter one is).
We plan to discuss current developments in this conjecture, in joint work with Dylan Thurston, as well as higher rank analogs of the conjecture that go well-beyond hyperbolic geometry.