Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Deletion-Contraction and the Surface Tutte Polynomial

Iain Moffatt, Maya Thompson

Abstract

In this paper we unify two families of topological Tutte polynomials. The first family is that coming from the surface Tutte polynomial, a polynomial that arises in the theory of local flows and tensions. The second family arises from the canonical Tutte polynomials of Hopf algebras. Each family includes the Las Vergnas, Bollobás-Riordan, and Krushkal polynomials. As a consequence we determine a deletion-contraction definition of the surface Tutte polynomial and recursion relations for the number of local flows and tensions in an embedded graph.

Deletion-Contraction and the Surface Tutte Polynomial

Abstract

In this paper we unify two families of topological Tutte polynomials. The first family is that coming from the surface Tutte polynomial, a polynomial that arises in the theory of local flows and tensions. The second family arises from the canonical Tutte polynomials of Hopf algebras. Each family includes the Las Vergnas, Bollobás-Riordan, and Krushkal polynomials. As a consequence we determine a deletion-contraction definition of the surface Tutte polynomial and recursion relations for the number of local flows and tensions in an embedded graph.
Paper Structure (8 sections, 11 theorems, 46 equations, 4 figures, 2 tables)

This paper contains 8 sections, 11 theorems, 46 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Packaged ribbon graphs
  3. A review of ribbon graphs
  4. Packaged ribbon graphs and operations on them
  5. A topological interpretation of packaged ribbon graphs
  6. A Tutte polynomial for packaged ribbon graphs
  7. Connections with other topological Tutte polynomials
  8. Recursively counting local flows and local tensions

Key Result

Proposition 2.7

Let $\underline{\mathbb{G}}=(\mathbb{G},\mathcal{V},\omega_{\mathcal{V}},\mathcal{F},\omega_{\mathcal{F}})$ be a packaged ribbon graph and $e\in E$ be an edge. Then

Figures (4)

  • Figure 1: A packaged ribbon graph $\underline{\mathbb{G}}$, its packagings $G(\mathbb{G};\mathcal{V})$ and $G(\mathbb{G}^*;\mathcal{F})$, and a subgraph $K$ of $G(\mathbb{G};\mathcal{V})$ with its corresponding ribbon subgraph $\mathbb{G}[K]$.
  • Figure 2: A graph embedded in a pseudo-surface and its corresponding packaged ribbon graph (weightings are omitted for clarity).
  • Figure 3: A computation of the packaged surface Tutte polynomial using deletion and contraction.
  • Figure 4: The relationship between $\boldsymbol{T}(\underline{\mathbb{G}};\boldsymbol{x},\boldsymbol{y})$ and other topological Tutte polynomials.

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Definition 2.8
  • Definition 3.1
  • ...and 24 more