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Graphs, Links, and Duality on Surfaces

Published online by Cambridge University Press:  29 September 2010

VYACHESLAV KRUSHKAL
VYACHESLAV KRUSHKAL*
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA (e-mail: krushkal@virginia.edu)
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Abstract

We introduce a polynomial invariant of graphs on surfaces, PG, generalizing the classical Tutte polynomial. Topological duality on surfaces gives rise to a natural duality result for PG, analogous to the duality for the Tutte polynomial of planar graphs. This property is important from the perspective of statistical mechanics, where the Tutte polynomial is known as the partition function of the Potts model. For ribbon graphs, PG specializes to the well-known Bollobás–Riordan polynomial, and in fact the two polynomials carry equivalent information in this context. Duality is also established for a multivariate version of the polynomial PG. We then consider a 2-variable version of the Jones polynomial for links in thickened surfaces, taking into account homological information on the surface. An analogue of Thistlethwaite's theorem is established for these generalized Jones and Tutte polynomials for virtual links.

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Combinatorics, Probability and Computing , Volume 20 , Issue 2 , March 2011 , pp. 267 - 287
Copyright
Copyright © Cambridge University Press 2010

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