Multivariable Tutte and transition polynomials
Seminar Room 1, Newton Institute
Tutte polynomial of graph theory is equivalent to the q-state Potts model partition function of statistical mechanics if the temperature and q are viewed as independent variables. Recent work (by, among many others, Sokal, who has done much to popularize the approach) has used a multivariable extension of the Potts model partition function, physically corresponding to allowing the interaction energies on the edges to vary. The Tutte polynomial was fully generalized by Zaslavsky, and, from a different point of view, Bollobas and Riordan, but the generalization requires care with a set of relations arising from three special small graphs. In 1987, Jaeger introduced transition polynomials of 4-regular graphs to unify polynomials given by vertex reconfigurations very similar to the skein relations of knot theory. These include the Martin polynomial (restricted to 4-regular graphs), the Kauffman bracket, and, for planar graphs via their medial graphs, the Penrose and classical Tutte polynomials. This talk discusses a generalized transition polynomial (developed jointly with I. Sarmiento) that extends the transition polynomials of Jaeger to arbitrary Eulerian graphs, and introduces pair weightings that function analogously to the parameterized edges in the generalized Tutte polynomial. The generalized transition polynomial and the generalized Tutte polynomial (and hence Potts model partition function) are related for planar graphs in much the same way as are Jaeger's transition polynomial and the classic Tutte polynomial. Moreover, the generalized transition polynomials are Hopf algebra maps. Thus, the comultiplication and antipode give recursive identities for generalized transition polynomials. A number of combinatorial identities then arise from these and the relations among these polynomials. Because of the medial graph construction relating these polynomials, the identities are relevant to the self-dual lattices that are 'natural'; underlying graphs of statistical mechanics.
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