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Constructing a Tutte polynomial for graphs embedded in surfaces

Iain Moffatt

TL;DR

This work constructs a unified Tutte polynomial for graphs embedded in surfaces, reconciling several existing topological extensions into a single invariant that coincides with the classical Tutte polynomial on plane graphs. It introduces three equivalent routes to define the polynomial—via a dichromatic-like Z with $Z(\mathbb{G};u,v)$, a state-sum with a modified rank $\rho$, and a deletion-contraction/duality framework—and demonstrates their equivalence, yielding the 2-variable Bollobás–Riordan (ribbon-graph) polynomial in this setting. A spanning quasi-tree expansion and a comprehensive discussion of properties (duality, universality, delta-matroid determination, and knot-polynomial connections) are provided. The results offer a principled foundation for topological graph polynomials, linking embedded-graph theory to knot theory and topological invariants while preserving reductions to the classical case when embedded in the plane.

Abstract

There are several different extensions of the Tutte polynomial to graphs embedded in surfaces. To help frame the different options, here we consider the problem of extending the Tutte polynomial to cellularly embedded graphs starting from first principles. We offer three different routes to defining such a polynomial and show that they all lead to the same polynomial. This resulting polynomial is known in the literature under a few different names including the ribbon graph polynomial, and 2-variable Bollobas-Riordan polynomial. Our overall aim here is to use this discussion as a mechanism for providing a gentle introduction to the topic of Tutte polynomials for graphs embedded in surfaces.

Constructing a Tutte polynomial for graphs embedded in surfaces

TL;DR

This work constructs a unified Tutte polynomial for graphs embedded in surfaces, reconciling several existing topological extensions into a single invariant that coincides with the classical Tutte polynomial on plane graphs. It introduces three equivalent routes to define the polynomial—via a dichromatic-like Z with , a state-sum with a modified rank , and a deletion-contraction/duality framework—and demonstrates their equivalence, yielding the 2-variable Bollobás–Riordan (ribbon-graph) polynomial in this setting. A spanning quasi-tree expansion and a comprehensive discussion of properties (duality, universality, delta-matroid determination, and knot-polynomial connections) are provided. The results offer a principled foundation for topological graph polynomials, linking embedded-graph theory to knot theory and topological invariants while preserving reductions to the classical case when embedded in the plane.

Abstract

There are several different extensions of the Tutte polynomial to graphs embedded in surfaces. To help frame the different options, here we consider the problem of extending the Tutte polynomial to cellularly embedded graphs starting from first principles. We offer three different routes to defining such a polynomial and show that they all lead to the same polynomial. This resulting polynomial is known in the literature under a few different names including the ribbon graph polynomial, and 2-variable Bollobas-Riordan polynomial. Our overall aim here is to use this discussion as a mechanism for providing a gentle introduction to the topic of Tutte polynomials for graphs embedded in surfaces.

Paper Structure

This paper contains 13 sections, 3 theorems, 39 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background on embedded graphs
  3. Graphs cellularly embedded in surfaces
  4. Ribbon graphs
  5. Further notation and terminology
  6. A Tutte polynomial for graphs embedded in surfaces
  7. A reminder of the classical Tutte polynomial of a graph
  8. Defining a polynomial through deletion-contraction relations
  9. Approach 1: Deducing a Tutte polynomial from $Z(\mathbb{G})$
  10. Approach 2: A state-sum approach to the Tutte polynomial of an embedded graph
  11. Approach 3: deletion-contraction, duality and the Tutte polynomial of an embedded graph
  12. Quasi-trees
  13. Further properties

Key Result

theorem 1

Let $\mathcal{G}$ be a minor-closed class of embedded graphs or ribbon graphs. Then there is a unique map $U: \mathcal{G}\rightarrow \mathbb{Z}[w, \sqrt{a^*}, \sqrt{c^*},\sqrt{a},\sqrt{c}]$ that satisfies where Moreover,

Figures (13)

  • Figure 1: Embedded graphs.
  • Figure 2: An instance of edge deletion.
  • Figure 3: An instance where edge deletion involves removing redundant handles.
  • Figure 4: An example where deletion and contraction changes the surface.
  • Figure 5: An instance of edge contraction.
  • ...and 8 more figures

Theorems & Definitions (8)

  • definition 1: Tutte polynomial --- first definition
  • definition 2: Tutte polynomial --- second definition
  • definition 3: Tutte polynomial --- third definition
  • theorem 1: Universality
  • theorem 2: Duality
  • proof
  • lemma 1
  • proof : Sketch