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

We develop an algebraic framework for ribbon graphs, revealing symmetry properties of (partial) twisted duality. The original ribbon group action of Ellis-Monaghan and Moffatt restricts self-duality, -petriality, or -triality to the canonical identification of a graph's edges with those of its dual, petrial, or trial, whereas the more natural definition allows any isomorphism. Here we define a new ribbon group action on ribbon graphs, using a semidirect product of the original ribbon group with a permutation group, to take (partial) twists and duals of ribbon graphs while also 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 exactly when it is isomorphic to one of its (partial) twisted duals. With these tools, we prove that every ribbon graph has in its orbit an orientable embedded bouquet, whose (partial) twisted duality properties propagate through the orbit. Thus, (partial) twisted duality properties of all embedded graphs may be analyzed through such bouquets, for which checking isomorphism reduces to checking just dihedral group symmetries. Previous research on self-duality, etc., typically focused on highly symmetric regular maps, but the theory here fully encompasses all cellularly embedded graphs. In contrast to the few, large, very high-genus, self-trial regular maps found by Wilson, and by Jones and Poultin, here we apply our framework to generate all self-trial ribbon graphs on up to seven edges. We also show how a graph's automorphism group may be used to find self-dual, etc., graphs in its orbit, thus exposing the relationship between regularity and the ribbon group action and, answering a question of Jones and Poulton, yielding an infinite family of self-trial graphs not arising as covers or parallel connections of regular maps.