[Paper Review] Topological Tutte Polynomial

Sergei Chmutov|arXiv (Cornell University)|Aug 27, 2017

Topological and Geometric Data Analysis41 references3 citations

TL;DR

This paper surveys topological extensions of the Tutte polynomial for graphs on surfaces, focusing on ribbon graphs and their generalizations. It establishes a unifying framework through the Krushkal polynomial, which generalizes the Bollobás-Riordan, Las Vergnas, and relative Tutte polynomials via quasi-tree expansions and duality relations, revealing deep connections across knot theory, matroid theory, and topological graph theory.

ABSTRACT

This is a survey recent works on topological extensions of the Tutte polynomial.

Motivation & Objective

• To unify various topological extensions of the Tutte polynomial—Bollobás-Riordan, Las Vergnas, and relative Tutte polynomials—within a single framework.
• To clarify the relationships between these polynomials through substitutions and reductions, particularly showing that Krushkal’s polynomial generalizes both the Bollobás-Riordan and Las Vergnas polynomials.
• To establish a quasi-tree expansion for the Krushkal polynomial, generalizing the classical spanning tree expansion of the Tutte polynomial.
• To explore duality and invariance properties, especially partial duality, in the context of virtual links and ribbon graphs.
• To suggest the existence of a more general matroid polynomial that unifies these topological extensions, based on emerging results in arithmetic and simplicial matroids.

Proposed method

• Utilizes ribbon graphs as a combinatorial model for cellularly embedded graphs on surfaces, representing topological embeddings via cyclic edge orderings and surface neighborhoods.
• Applies Clark Butler’s quasi-tree expansion to express the Krushkal polynomial as a sum over spanning quasi-trees, with contributions from the abstract graph and surface topology.
• Employs substitutions to relate the Krushkal polynomial to other known polynomials: setting $ Z = 1 $ recovers the Las Vergnas polynomial, while $ X = x-1, Y = y-1, Z = (XY)^{-1/2} $ yields the Bollobás-Riordan polynomial.
• Uses partial duality (generalized duality) to explain invariance of certain specializations of the Bollobás-Riordan polynomial under duality transformations in virtual link theory.
• Analyzes the contribution of each quasi-tree via topological invariants: $ A^{s(F(Q))/2} $ and $ B^{s(F(Q^*)) / 2} $, where $ s $ is the number of boundary components of the ribbon graph.
• Demonstrates that the Krushkal polynomial reduces to the relative Tutte polynomial under specific substitutions, and that the Las Vergnas and Bollobás-Riordan polynomials are independent.

Experimental results

Research questions

• RQ1 How can the Tutte polynomial be generalized to graphs embedded on surfaces, particularly in non-planar and non-orientable cases?
• RQ2 What is the precise relationship between the Krushkal polynomial, the Bollobás-Riordan polynomial, and the Las Vergnas polynomial, and how do they relate via substitutions?
• RQ3 Can a unified quasi-tree expansion be established for the Krushkal polynomial that generalizes the classical Tutte polynomial's spanning tree expansion?
• RQ4 What role does partial duality play in explaining invariance properties of link polynomials such as the Jones polynomial in virtual knot theory?
• RQ5 Is there a more general matroid polynomial that unifies the Krushkal, Bollobás-Riordan, and Las Vergnas polynomials, and how might it relate to arithmetic or simplicial matroids?

Key findings

• The Krushkal polynomial generalizes both the Bollobás-Riordan and Las Vergnas polynomials via distinct substitutions: setting $ X = x-1, Y = y-1, Z = 1 $ recovers the Las Vergnas polynomial, while $ X = x-1, Y = y-1, Z = (XY)^{-1/2} $ yields the Bollobás-Riordan polynomial.
• The Krushkal polynomial admits a quasi-tree expansion, as shown by Clark Butler, which generalizes the classical Tutte polynomial’s spanning tree expansion to topological settings.
• For the example ribbon graph with four edges, the Krushkal polynomial evaluates to $ K_G = (X+1)(Y+1)B + (Y+1)B + (Y+1)A^{1/2}B^{1/2} + (X+A+2)(Y+1) $, confirming the expansion via explicit computation.
• The polynomial is invariant under partial duality, which explains the invariance of certain specializations of the Bollobás-Riordan polynomial in virtual link invariants.
• The relative Tutte polynomial and the Las Vergnas polynomial are independent, suggesting that a more general matroid polynomial may unify these extensions.
• The Krushkal polynomial also reduces to the relative Tutte polynomial under a specific substitution, and its equivalence to the generalized duality framework in virtual links has been clarified.