Combinatorics Seminar- Searching for Trialities of Embedded Graphs

Speaker: Lowell Abrams, GWU
Date and time: Thursday, November 1, 4-5pm
Place:  Phillips 110
Title: Searching for Trialities of Embedded Graphs

Abstract: The actions of dualizing and Petrie-dualizing carry one isomorphism class of graph embeddings to another. These operations generate a group isomorphic to S_3 called the Wilson group, and the order-3 elements in the Wilson group are called trialities. Historically, it has been hard to find graph embeddings that are self-trial -- fixed by the trialities -- but not also self-dual and self-Petrie-dual.

In this talk, I will describe the theoretical framework Jo Ellis-Monaghan (St. Michael's College) and I developed to tackle the problem of finding graph embeddings of this elusive type, as well as a variety of bijections and construction techniques we used in conjunction with the theoretical framework to successfully implement a computer search to find all self-trial but non-self-dual and non-self-Petrie-dual embeddings with up to seven edges. As far as we can tell, none of these has ever been found before.