10/21/2004

Stavros Garoufalidis
Georgia Tech.

The Geometry of the Jones polynomial

The Jones polynomial of a knot in 3-space is a powerful quantum field theory invariant.The Jones polynomial is a Laurent polynomial, and it can be enhanced to a sequence of Laurent polynomials. This sequence is not random. Instead, we will show that this sequence is q-holonomic, ie that it satisfies a recursion relation. This phenomenon can be extended to links, and to quantum invariants of higher rank Lie groups. We will show from first principles that holonomicity is a general property of statistical mechanics models. Using holonomicity, and specializing to q=1, allows us to define a 'characteristic variety of a knot', which in the SL_2 case is a complex curve in C^2. We conjecture that the characteristic variety of a knot coincides with its deformation variety. We give evidence for the 'characteristic equals deformation variety' conjecture. Time permitting, we plan to discuss briefly the implications of holonomicity to the hyperbolic volume conjecture.