On a conjecture of Gross, Mansour and Tucker for $\Delta$-matroids

TL;DR

This paper investigates the properties of the partial-duality polynomial for delta-matroids, extending previous work on ribbon graphs and binary delta-matroids, and finds that even non-binary delta-matroids do not exhibit width-changing twists.

Contribution

It generalizes the study of partial-duality polynomials to even non-binary delta-matroids, showing they lack width-changing twists, thus broadening understanding of delta-matroid properties.

Findings

No even non-binary delta-matroids have width-changing twists.

Extends previous results from binary to non-binary delta-matroids.

Supports conjectures about the behavior of partial-duality polynomials.

Abstract

Gross, Mansour, and Tucker introduced the partial-duality polynomial of a ribbon graph [Distributions, European J. Combin. 86, 1--20, 2020], the generating function enumerating partial duals by the Euler genus. Chmutov and Vignes-Tourneret wondered if this polynomial and its conjectured properties would hold for general delta-matroids, which are combinatorial abstractions of ribbon graphs. Yan and Jin contributed to this inquiry by identifying a subset of delta-matroids-specifically, even normal binary ones-whose twist polynomials are characterized by a singular term. Building upon this foundation, the current paper expands the scope of the investigation to encompass even non-binary delta-matroids, revealing that none of them have width-changing twists.