[Paper Review] A quasi-tree expansion of the Krushkal polynomial

Clark Butler|arXiv (Cornell University)|May 2, 2012

Geometric and Algebraic Topology6 references4 citations

TL;DR

This paper introduces a generalized Krushkal polynomial for graphs embedded in compact, possibly non-orientable surfaces, proving it admits a natural quasi-tree expansion. The key contribution is that this expansion unifies and extends previous quasi-tree expansions of the Bollobás-Riordan and Las Vergnas polynomials, establishing a comprehensive framework for surface-embedded graph invariants via quasi-trees.

ABSTRACT

We introduce a generalization of the Krushkal polynomial to nonorientable surfaces, and prove that this polynomial has a natural quasi-tree expansion. This generalized Krushkal polynomial contains the Bollobás-Riordan polynomial of a (possibly nonorientable) ribbon graph as a specialization. The quasi-tree expansion proven here then extends the recent quasi-tree expansions of the Bollobás-Riordan polynomial deduced in the oriented case by A. Champanerkar et al. and in the more general unoriented case by E. Dewey and F. Vignes-Tourneret. The generalized Krushkal polynomial also contains the Las Vergnas polynomial of a cellulation of a surface as a specialization; we use this fact to deduce a quasi-tree expansion for the Las Vergnas polynomial.

Motivation & Objective

• To extend the Krushkal polynomial to graphs embedded in non-orientable surfaces, preserving its duality and topological properties.
• To prove that this generalized polynomial admits a quasi-tree expansion, extending prior results from the oriented case.
• To show that the generalized Krushkal polynomial specializes to both the Bollobás-Riordan and Las Vergnas polynomials.
• To derive a new quasi-tree expansion for the Las Vergnas polynomial using the generalized framework.

Proposed method

• Introduces a generalized Krushkal polynomial using topological invariants: component count, kernel dimension of homology, and genus of surface neighborhoods.
• Defines the polynomial as a sum over all spanning subgraphs, with monomials built from topological invariants: c(F)−c(G), k(F), s(F)/2, and s⊥(F)/2.
• Applies a total edge ordering to ribbon graphs to define quasi-trees as spanning subgraphs with a single face.
• Uses Poincaré duality and matroid-theoretic duality to relate the generalized Krushkal polynomial to the Las Vergnas polynomial via specialization.
• Proves the quasi-tree expansion by leveraging the duality of the generalized Krushkal polynomial and the structure of quasi-tree decompositions.
• Establishes the identity LV_{G,Σ}(X,Y,Z) = Z^{s(Σ)/2} P_{G,Σ}(X−1,Y−1,Z^{−1},Z) to link the Las Vergnas polynomial to the generalized Krushkal polynomial.

Experimental results

Research questions

• RQ1 Can the Krushkal polynomial be generalized to non-orientable surfaces while preserving its duality and topological invariants?
• RQ2 Does the generalized Krushkal polynomial admit a quasi-tree expansion analogous to the Bollobás-Riordan polynomial in the non-orientable case?
• RQ3 How does the generalized Krushkal polynomial relate to the Las Vergnas polynomial of a cellularly embedded graph?
• RQ4 Can the quasi-tree expansion of the Las Vergnas polynomial be derived from the generalized Krushkal polynomial via specialization?
• RQ5 What is the precise relationship between the topological invariants in the generalized Krushkal polynomial and the matroid-theoretic invariants in the Las Vergnas polynomial?

Key findings

• The generalized Krushkal polynomial is defined for graphs embedded in any compact surface, including non-orientable ones, and retains the duality property P_{G,Σ}(X,Y,A,B) = P_{G*,Σ}(Y,X,B,A).
• The polynomial admits a quasi-tree expansion: P_{G,Σ}(X,Y,A,B) = ∑_{Q∈Q_G} T_{G_Q}(X,A) T_{G*_Q^*}(Y,B) A^{n(R_{VE(Q)}) - n(G_Q)} B^{n(R_{VE(Q)}) - n(G_Q)}
• The generalized Krushkal polynomial specializes to the Bollobás-Riordan polynomial for any ribbon graph, extending prior quasi-tree expansions to the non-orientable case.
• The generalized Krushkal polynomial also specializes to the Las Vergnas polynomial via the identity LV_{G,Σ}(X,Y,Z) = Z^{s(Σ)/2} P_{G,Σ}(X−1,Y−1,Z^{−1},Z).
• A new quasi-tree expansion for the Las Vergnas polynomial is derived: LV_{G,Σ}(X,Y,Z) = ∑_{Q∈Q_G} T_{G_Q}(X,Z^{−1}) T_{G*_Q^*}(Y,Z) Z^{n(R_{VE(Q)}) - n(G_Q)}
• The proof relies on identifying the dual of the quasi-tree's vertex-edge subcomplex and using genus and component count identities to match monomial exponents.