Title: Jones Polynomial, the Potts Model and Khovanov Homology
Speaker: Louis H. Kauffman, UIC
Abstract: The Jones polynomial invariant in knot theory and the Potts
model in statistical mechanics are closely related through the bracket
state sum model -- a partition function defined on knot diagrams that
specializes to the Jones polynomial and can, by different specialization,
represent the dichromatic and Tutte polynomials for plane graphs. Via this
connection, one can use knot and link diagrams to represent the partition
function for the Potts model. The loops in the bracket expansion then
correspond to boundaries of regions of constant spin in the Potts model.
These states (loop collections)  in the bracket model are elevated to a
category whose homology is Khovanov homology, an invariant more powerful
than the Jones polynomial. From the point of view of the physics of the
Potts model it is natural to ask for a physical interpretation of this
homology theory based on states delineating regions of constant spin. We
will raise these questions and discuss how Khovanov homology and its
graded Euler characteristic look from the point of view of the Potts
model. We will also point out how this way of thinking leads to a
quantum-information theoretic reformulation of Khovanov homology and the
Jones polynomial.