Topological graph theory
On the evaluation at (3,3) of the Tutte polynomial of a graph
Journal of Combinatorial Theory Series B
Complexity: knots, colourings and counting
Complexity: knots, colourings and counting
European Journal of Combinatorics - In memoriam François Jaeger
Evaluations of the circuit partition polynomial
Journal of Combinatorial Theory Series B
Knot invariants and the Bollobás-Riordan polynomial of embedded graphs
European Journal of Combinatorics
Generalized duality for graphs on surfaces and the signed Bollobás--Riordan polynomial
Journal of Combinatorial Theory Series B
Graphs, links, and duality on surfaces
Combinatorics, Probability and Computing
A Penrose polynomial for embedded graphs
European Journal of Combinatorics
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In [2,3], Bollobas 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). We then relate R(G) to the generalized transition polynomial 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) 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) and Q(G).