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

Charlton Li

TL;DR

The paper defines and analyzes the twist polynomial $ ext{ obracket} ext{ ext{$w$}}_{D}(z)$ for delta-matroids and its relation to the partial-dual Euler-genus polynomial of ribbon graphs. It establishes a rank-based framework for looped simple graphs, linking $w(D(G)*A)$ to principal submatrix ranks of the adjacency matrix, and proves a recursive formula for the twist polynomial under the one-point join of two looped simple graphs, with explicit auxiliary polynomials $Q_1,Q_2,Q_3$ guiding the combination. Special evaluations at $z=-\tfrac{1}{2}$ yield linear relations among twist polynomials under loop complementation, and a simpler unlooped case recovers known recurrences. The authors also give a delta-matroid–only description of the one-point join, via a determinant identity, and prove that Yan–Jin’s leaf recursion generalizes to delta-matroids, thereby extending the twist polynomial framework beyond signed intersection graphs. Together, these results unify ribbon-graph dualities, delta-matroid twists, and join operations, with potential extensions to vf-safe delta-matroids and topological interpretations of the recursion.

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-Poincaré 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.

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

TL;DR

The paper defines and analyzes the twist polynomial w for delta-matroids and its relation to the partial-dual Euler-genus polynomial of ribbon graphs. It establishes a rank-based framework for looped simple graphs, linking to principal submatrix ranks of the adjacency matrix, and proves a recursive formula for the twist polynomial under the one-point join of two looped simple graphs, with explicit auxiliary polynomials guiding the combination. Special evaluations at yield linear relations among twist polynomials under loop complementation, and a simpler unlooped case recovers known recurrences. The authors also give a delta-matroid–only description of the one-point join, via a determinant identity, and prove that Yan–Jin’s leaf recursion generalizes to delta-matroids, thereby extending the twist polynomial framework beyond signed intersection graphs. Together, these results unify ribbon-graph dualities, delta-matroid twists, and join operations, with potential extensions to vf-safe delta-matroids and topological interpretations of the recursion.

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-Poincaré 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 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.

Paper Structure

This paper contains 10 sections, 26 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Ribbon Graphs
  4. Delta-Matroids
  5. Ribbon-Graphic and Binary Delta-Matroids
  6. A Rank Formula for the Twist Polynomial on Looped Simple Graphs
  7. The Twist Polynomial of a One-Point Join of Graphs
  8. The Delta-Matroid Associated to a One-point Join of Graphs
  9. Leaf Recursion for Delta-Matroids
  10. Concluding Remarks

Key Result

Proposition 1

(Theorem 4.1 in Yan_Jin_2022). If two bouquets $B_1$ and $B_2$ have isomorphic signed intersection graphs, then $\prescript{\partial}{}{\varepsilon_{B_1}}(z)=\prescript{\partial}{}{\varepsilon_{B_2}}(z).$

Theorems & Definitions (43)

  • Proposition 1
  • Proposition 2
  • Definition 3
  • Theorem 4
  • Theorem 5
  • Lemma 6
  • Lemma 7
  • Proposition 8
  • Lemma 9
  • Lemma 10
  • ...and 33 more