|
Organizers |
State Models and the Jones Polynomial
by
Louis H. Kauffman
University of Illinois at Chicago
This talk will begin with a recap of the construction of the (Kauffman) bracket state
summation for the Jones polynomial and its relationship with the diagrammatic
Temperley-Lieb algebra and hence with the original representation of the braid group
to that algebra by Vaughan Jones. We then discuss how the bracket state sum is a
special case of the formalism for the partition function
for the Potts model in statistical mechanics. We show how the bracket state sum
fits into the framework of other invariants and into the framework of the
Yang-Baxter equation.We then keep looking at the idea of combinatorial state sums
with forays into the Alexander polynomial, the arrow polynomial in
virtual knot theory and formulations of Khovanov
homology that are closely related to state summations. This talk has a semi-historical form,
but it is intended to explore the possibilities in these constructions that we understand only
a little bit at the present time.
Date received: May 5, 2014
Copyright © 2014 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbjb-05.