Sponsored by the Rutgers University Department of Mathematics and the

Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

**Co-organizers:****Andrew Baxter**, Rutgers University, baxter{at} math [dot] rutgers [dot] edu**Doron Zeilberger**, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu

Title: Quantum spin network evaluations

Speaker: **Roland van der Veen**, University of Amsterdam

Date: Thursday, November 5, 2009 5:00pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

In this talk we introduce spin networks, which are labeled graphs representing contractions of SU(2) invariant tensors, widely used in quantum theory, statistical mechanics and knot theory. We will show how to evaluate such networks in terms of hypergeometric multisums and how these sums can be encoded neatly in terms of an explicit rational generating function defined in terms of the network.

In knot theory one is led to study q-analogues of these spin networks, whose evaluations are q-hypergeometric multisums closely related to the Jones polynomial. We'll sketch how this works and how one can in many cases guess the correct q-evaluation of the spin network by looking at the classical formulas in the "right" way. Making this precise is a challenge for experimental mathematics since in some cases the guessed answers are known to be false. In the case of planar spin networks we will discuss how to resolve the matter in terms of a modified generating function.