Medial graphs and twisted duals

Iain Moffatt

University of South Alabama

Abstract

We consider two operations on the edge of an embedded graph (or ribbon graph): giving a half-twist to the edge and taking the partial dual with respect to the edge. These two operations give rise to an action of S_3^{e(G)}, the "ribbon group" of G, on G. We show that this ribbon group action gives a complete characterization of duality in that if G is any cellularly embedded graph with medial graph G_m, then the orbit of G under the group action is precisely the set of all graphs with medial graphs isomorphic (as abstract graphs) to G_m. If there is time, I may also mention some of the applications to graph polynomials.