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

In [2] and [3], Bollobás and Riordan (2001, 2002) 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’ restating the universality property of this topological Tutte polynomial, R(G)R(G). We then relate R(G)R(G) to the generalized transition polynomial Q(G)Q(G) of Ellis-Monaghan and Sarmiento (2002) [18] 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)R(G) that holds for both orientable and unorientable ribbon graphs. We conclude by placing the results of Chumutov and Pak (2007) [11] for virtual links in the context of the relation between R(G)R(G) and Q(G)Q(G).