Twisted duality for embedded graphs

TL;DR

This paper introduces a group action on embedded graphs using edge operations, providing a comprehensive framework for understanding dualities and their relation to graph polynomials.

Contribution

It extends duality concepts through the ribbon group action, characterizes duals via medial graphs, and links these to graph polynomials.

Findings

The ribbon group action characterizes all graphs with a given medial graph.

Partial duals can be characterized in terms of medial graphs.

The generalized transition polynomial interacts with the ribbon group action.

Abstract

We consider two operations on an edge of an embedded graph (or equivalently a 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, on G. The action of the ribbon group on embedded graphs extends the concepts of duality, partial duality and Petrie duality. 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. We provide characterizations of special sets of twisted duals, such as the partial duals, of embedded graphs in terms of medial graphs and we show how different kinds of graph isomorphism give rise to these various notions of duality. The ribbon group action then leads to a deeper understanding of the properties of, and relationships among, various graph polynomials via the generalized transition polynomial which interacts naturally with the ribbon group action.