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

Lowell Abrams, Jo Ellis-Monaghan

Published 2019-01-11Version 1

We present twisted duality tools, in particular the \emph{ribbon group action}, to identify and generate new graphs with various forms of self-duality including geometric duality, Petrie duality, and Wilson duality, as well as triality. Previous work typically focused on regular maps, but the methods presented here apply to general embedded graphs. We use partial duals and twists to reduce the search for graphs with some form of self-duality/triality to the study of one vertex ribbon graphs. The orbits of these one vertex graphs partition all embedded graphs, and we show that the twisted dualities and trialities of the one vertex graphs propagate through their orbits. This leads to a solution process for determining all types of self-duality/triality for general embedded graphs. The same methods also apply to discover graphs with desired twisted duality properties in the orbits of graphs with rich automorphism groups. We provide an algorithm that will find all graphs with any of the various forms of self-duality/triality in the orbit of a graph that is isomorphic to any twisted dual or trial of itself, and give examples generated by its implementation. As an application of the results, we find numerous self-trial graphs of low genus on very few edges, in contrast to the very large, high genus regular map examples of Wilson (\emph{Operators over regular maps}), and Jones and Poultin (\emph{Maps admitting trialities but not dualities}).