On a conjecture of Gross, Mansour and Tucker

TL;DR

This paper investigates a conjecture about partial duality in ribbon graphs, proving that the known counterexamples are essentially the only exceptions to the conjecture that partial duals can change genus.

Contribution

It proves that the identified counterexamples are the only ones, confirming the conjecture's validity beyond these cases.

Findings

Counterexamples are essentially the only exceptions

Partial duality can change genus in all other cases

Confirms the conjecture for all but known counterexamples

Abstract

Partial duality is a duality of ribbon graphs relative to a subset of their edges generalizing the classical Euler-Poincare duality. This operation often changes the genus. Recently J.L.Gross, T.Mansour, and T.W.Tucker formulated a conjecture that for any ribbon graph different from plane trees and their partial duals, there is a subset of edges partial duality relative to which does change the genus. A family of counterexamples was found by Qi Yan and Xian'an Jin. In this note we prove that essentially these are the only counterexamples.