Graphs, links, and duality on surfaces

TL;DR

This paper introduces a new polynomial invariant for graphs on surfaces that generalizes the Tutte polynomial, explores its duality properties, and relates it to the Jones polynomial for links in thickened surfaces.

Contribution

It defines a polynomial invariant $P_G$ for graphs on surfaces, establishes its duality properties, and connects it to the Bollobás-Riordan polynomial and generalized Jones polynomial for links.

Findings

$P_G$ generalizes the Tutte polynomial for surface graphs.

Duality properties of $P_G$ mirror classical planar duality.

Relation between $P_G$ and Bollobás-Riordan polynomial established.

Abstract

We introduce a polynomial invariant of graphs on surfaces, $P_G$, generalizing the classical Tutte polynomial. Topological duality on surfaces gives rise to a natural duality result for $P_G$, 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, $P_G$ specializes to the well-known Bollobas-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 $P_G$. 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.