A recursion for the twist polynomial of a one-point join of normal binary delta-matroids

TL;DR

This paper derives a recursion formula for the twist polynomial of one-point joins of normal binary delta-matroids, generalizing previous results and providing new relations and characterizations in graph and delta-matroid theory.

Contribution

It introduces a new recursion for the twist polynomial of one-point joins of delta-matroids, extending prior work on partial-dual Euler-genus polynomials.

Findings

Recursion formula for the twist polynomial of one-point joins

Relations for the twist polynomial evaluated at -1/2

Characterization of feasible sets in delta-matroids

Abstract

The partial-dual Euler-genus polynomial was defined by Gross, Mansour, and Tucker to analyze how the Euler genus of a ribbon graph changes under partial duality, a generalization of Euler-Poincar\'{e} duality introduced by Chmutov. The twist polynomial defined by Yan and Jin extends the partial-dual Euler-genus polynomial to a polynomial on delta-matroids. We derive a recursion formula for the twist polynomial of a one-point join of looped simple graphs -- equivalently, normal, binary delta-matroids. Our recursion applies to the partial-dual Euler-genus polynomial as a special case, where it generalizes a recursion obtained by Yan and Jin. We obtain relations for the twist polynomial on looped simple graphs evaluated at $-1/2$ and for the twist polynomial of a graph with a single looped vertex. A characterization is given for the feasible sets of the delta-matroid associated to a one-point join of looped simple graphs. We show that Yan and Jin's recursion extends to the twist polynomial on delta-matroids.