What is a holonomic function? (or Zeilberger meets Jones and Thurston)

A function of several variables is called holonomic if it satisfies a maximally overdetermined system of linear differential equations with polynomial coefficients. Zeilberger noticed that the abstract notion of holonomicity can be applied to verify, in a systematic and computerized way, combinatorial identities among special functions.

Using a general state sum definition of the colored Jones function of a link in 3-space, we prove from first principles that the colored Jones function is a multisum of q-proper-hypergeometric function, and thus it is q-holonomic.

This explains the "Zeilberger meets Jones" part of the title.

Specializing the recursion relation at q=1 gives rise to a complex curve associated to a knot, which we conjecture that it is identified with the complex curve of SL(2,C) representations of the knot complement, viewed from the boundary. We prove this conjecture for the trefoil and figure 8 knots.

This explains the "Zeilberger meets Thurston" part of the title.

The recursion relations for the colored Jones function are a puzzle to current physics. They ought to be explained via new invariants of links discovered by M-theory (under the name: BPS states). Permitting time, I'll discuss this speculation a bit.

The talk is on joint work with Thang Le, which can be found on the arXiv math.GT/0309214 and math.GT/0306230.