Twist polynomial interpolation for binary delta-matroids

TL;DR

This paper characterizes the parity properties of the twist polynomial for binary delta-matroids, extending the understanding of partial-dual polynomials in ribbon graph theory.

Contribution

It proves that the twist polynomial of any binary delta-matroid is either even, odd, or both, answering an open question about partial-dual polynomials.

Findings

The twist polynomial of binary delta-matroids is classified as even, odd, or both.

This classification applies to partial-dual polynomials of ribbon graphs.

The results resolve an open problem posed by Gross, Mansour, and Tucker.

Abstract

Gross, Mansour and Tucker introduced the partial-dual polynomial of a ribbon graph and asked under what conditions such a polynomial is even-interpolating, odd-interpolating, or both. In this paper, we provide an answer to this open problem.Using the framework of delta-matroids, we prove that the twist polynomial of any binary delta-matroid is either an even polynomial, an odd polynomial, or both even-interpolating and odd-interpolating. Applying this to ribbon graphs, we deduce that the partial-dual polynomial of any ribbon graph satisfies the same conclusion.