Stavros Garoufalidis

Abstract:   The Jones polynomial of a knot in 3-space is a fascinating polynomial invariant that encodes the noncommutativity of the over-under crossings of a knot. The Jones polynomial may be defined using statistical mechanics, path integrals, quantum groups or other suitable noncommutative tools. Despite the numerous definitions, the geometry and topology of the Jones polynomial is tighly hidden.

In our talk we reveal a connection between the Jones polynomial, and holonomic functions. The latter are functions which satisfy a maximum number of independent different/differential equations. The connection comes from first principles, and reveals that the Jones polynomial is given by a multisum of q-hypergeometric functions.

Specializing the difference equations at q = 1 gives rise to a complex curve associated to a knot.

It is conjectured (and verified for the simplest knots) that this complex curve coincides with the curve of deformations of SL(2,C) representations of the knot complement, viewed from the boundary.

There is some speculation about the physical origin of the recursion relations, and what else they may teach us about the geometry of the Jones polynomial.