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

In [BR01], [BR02], Bollobás 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) of [E-MS02] 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 [CP07] for virtual links in the context of the relation between R(G) and Q(R). ∗ Department of Mathematics, Saint Michael’s College, 1 Winooski Park, Colchester, VT 05439. [email protected] ∗∗ Department of Mathematics, Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, I-00133, Rome, Italy. [email protected] 1 Support was provided by the National Security Agency and by the Vermont Genetics Network through Grant Number P20 RR16462 from the INBRE Program of the National Center for Research Resources (NCRR), a component of the National Institutes of Health (NIH). This paper’s contents are solely the responsibility of the authors and do not necessarily represent the official views of NCRR or NIH.