Topological graph theory
New results for the Martin polynomial
Journal of Combinatorial Theory Series B
On the number of 3-edge colorings of cubic graphs
European Journal of Combinatorics
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
A recipe theorem for the topological Tutte polynomial of Bollobás and Riordan
European Journal of Combinatorics
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We extend the Penrose polynomial, originally defined only for plane graphs, to graphs embedded in arbitrary surfaces. Considering this Penrose polynomial of embedded graphs leads to new identities and relations for the Penrose polynomial which cannot be realized within the class of plane graphs. In particular, by exploiting connections with the transition polynomial and the ribbon group action, we find a deletion-contraction-type relation for the Penrose polynomial. We relate the Penrose polynomial of an orientable chequerboard colourable graph to the circuit partition polynomial of its medial graph and use this to find new combinatorial interpretations of the Penrose polynomial. We also show that the Penrose polynomial of a plane graph G can be expressed as a sum of chromatic polynomials of twisted duals of G. This allows us to obtain a new reformulation of the Four Colour Theorem.