New Dualities From Old: generating geometric, Petrie, and Wilson dualities and trialites of ribbon graphs

Abstract

We develop an algebraic framework for ribbon graphs that reveals symmetry properties of (partial) twisted duality. The original ribbon group action of Ellis-Monaghan and Moffatt restricts self-duality, self-petriality, or self-triality to the canonical identification between the edges of a graph and those of its dual, petrial, or trial, whereas the more natural usual definition allows any isomorphism. Here we define a new ribbon group action on ribbon graphs that uses a semidirect product of the original ribbon group with a permutation group to take (partial) twists and duals of ribbon graphs while simultaneously encoding graph isomorphisms. This brings new algebraic tools to bear on the natural definitions of self-duality etc., as a ribbon graph is a fixed point of this new ribbon group action if and only if it is isomorphic to one of its (partial) twisted duals. Using these new tools, we show that every ribbon graph has in its orbit an orientable embedded bouquet and prove that the (partial) twisted duality properties of these bouquets propagate through their orbits. Thus, all (partial) twisted duality properties of general embedded graphs, especially those of being self-dual, self-petrial, and self-trial, may be analyzed through orientable embedded bouquets, for which checking isomorphism reduces simply to checking dihedral group symmetries. Previous research on self-duality, self-petriality, and self-triality typically focused on highly symmetric regular maps. However, the theory here fully encompasses all cellularly embedded graphs. In contrast with the few, large, very high-genus, self-trial regular maps found by Wilson, and by Jones and Poultin, here we apply the new ribbon group action to generate all self-trial ribbon graphs on up to seven edges. We also show how the automorphism group of a graph may be used to find self-dual, -petrial or trial graphs in its orbit, thus exposing the relationship between regularity and the ribbon group action. This strategy yields an infinite family of self-trial graphs on 3m edges, for all m, that do not arise as covers or parallel connections of regular maps, thus answering a question of Jones and Poulton.