A recipe theorem for the topological Tutte polynomial of Bollobas and Riordan

TL;DR

This paper develops a recipe theorem for the topological Tutte polynomial of ribbon graphs, relating it to the transition polynomial and establishing duality properties, thereby extending classical graph polynomial relations to topological surfaces.

Contribution

It provides a new recipe theorem for the Bollobás-Riordan topological Tutte polynomial and links it to the transition polynomial, extending classical relations to ribbon graphs.

Findings

Established a recipe theorem for R(G)

Connected R(G) to the transition polynomial Q(G)

Proved a duality property for R(G)

Abstract

In [A polynomial invariant of graphs on orientable surfaces, Proc. Lond. Math. Soc., III Ser. 83, No. 3, 513-531 (2001)] and [A polynomial of graphs on surfaces, Math. Ann. 323, 81-96 (2002)], Bollobas and Riordan generalized the classical Tutte polynomial to graphs cellularly embedded in surfaces, i.e. ribbon graphs, thus encoding topological information not captured by the classical Tutte polynomial. We provide a `recipe theorem' for their new topological Tutte polynomial, R(G). We then relate R(G) to the generalized transition polynomial Q(G) via a medial graph construction, thus extending the relation between the classical Tutte polynomial and the Martin, or circuit partition, polynomial to ribbon graphs. We use this relation to prove a duality property for R(G) that holds for both oriented and unoriented ribbon graphs. We conclude by placing the results of Chumutov and Pak [The Kauffman bracket and the Bollobas-Riordan polynomial of ribbon graphs, Moscow Mathematical Journal 7(3) (2007) 409-418] for virtual links in the context of the relation between R(G) and Q(R).