  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).
