The Structure of the Tutte–Grothendieck Ring of Ribbon Graphs

Tutte Polynomials of Flower Graphs

A Motivic Approach to Phase Transitions in Potts Models

We describe an approach to the study of phase transitions in Potts models based on an estimate of the complexity of the locus of real zeros of the partition function, computed in terms of the classes in the Grothendieck ring of the affine algebraic varieties defined by the vanishing of the multivariate Tutte polynomial. We give completely explicit calculations for the examples of the chains of ...

Quasi-tree expansion for the Bollobás--Riordan--Tutte polynomial

Bollobás and Riordan introduced a three-variable polynomial extending the Tutte polynomial to oriented ribbon graphs, which are multi-graphs embedded in oriented surfaces, such that complementary regions (faces) are discs. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. By generalizing Tutte’s concept of activity to quasi-tree...

The Tutte Dichromate and Whitney Homology of Matroids

We consider a specialization YM (q, t) of the Tutte polynomial of a matroid M which is inspired by analogy with the Potts model from statistical mechanics. The only information lost in this specialization is the number of loops of M . We show that the coefficients of YM (1 − p, t) are very simply related to the ranks of the Whitney homology groups of the opposite partial orders of the independe...

Strong Tutte Functions of Matroids and Graphs

A strong Tutte function of matroids is a function of finite matroids which satisfies F ( M 1$M2) = F ( M 1 ) F ( M 2 ) and F ( M ) = aeF(M\e) + b e F ( M / e ) for e not a loop or coloop of M ,where ae , be are scalar parameters depending only on e . We classify strong Tutte functions of all matroids into seven types, generalizing Brylawski's classification of Tutte-Grothendieck invariants. One...