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Quantum state systems that count perfect matchings

Scott Baldridge, Ben McCarty

TL;DR

The paper advances a categorification program for Penrose-inspired colorings by constructing bigraded $n$-color vertex homology and a filtered variant via a spectral sequence, with the graded Euler characteristic recovering the $n$-color vertex polynomial $\llangle \Gamma\rrangle_{n}$. It establishes a deep link between quantum-state–style constructions and combinatorial face colorings on ribbon graphs, enabling, in the $n=2$ case, a homological count of perfect matchings and a broader interpretation in terms of partial $n$-face colorings. The vertex polynomial $V(\Gamma,n)$ and the total matching polynomial $TM(\Gamma,n)$ emerge as convenient encodings of these homological invariants, and the theory extends to $4$-regular graphs with analogous structures. Overall, the work provides a robust, algebraic framework for understanding colorings of ribbon graphs, connecting Penrose-type formulas to topological quantum field theory and suggesting avenues toward new graph and manifold invariants. The constructions yield computable invariants and open pathways to relate colorings to harmonic/color-based decompositions and potential spin-network–style interpretations in topology.

Abstract

In this paper we show how to categorify the $n$-color vertex polynomial, which is based upon one of Roger Penrose's formulas for counting the number of $3$-edge colorings of a planar trivalent graph. Using topological quantum field theory (TQFT), we introduce a quantum state system to build a new bigraded theory called the bigraded $n$-color vertex homology. The graded Euler characteristic of this homology is the $n$-color vertex polynomial. We then produce a spectral sequence whose $E_\infty$-page is a filtered theory called filtered $n$-color vertex homology and show that it is generated by certain types of face colorings of ribbon graphs. For $n=2$, we show that the filtered $n$-color vertex homology is generated by face colorings that correspond to perfect matchings. Finally, we introduce and give meaning to what the vertex polynomial counts when $n \geq 2$. This polynomial is a new abstract graph invariant that can be inferred from certain formulas of Penrose.

Quantum state systems that count perfect matchings

TL;DR

The paper advances a categorification program for Penrose-inspired colorings by constructing bigraded -color vertex homology and a filtered variant via a spectral sequence, with the graded Euler characteristic recovering the -color vertex polynomial . It establishes a deep link between quantum-state–style constructions and combinatorial face colorings on ribbon graphs, enabling, in the case, a homological count of perfect matchings and a broader interpretation in terms of partial -face colorings. The vertex polynomial and the total matching polynomial emerge as convenient encodings of these homological invariants, and the theory extends to -regular graphs with analogous structures. Overall, the work provides a robust, algebraic framework for understanding colorings of ribbon graphs, connecting Penrose-type formulas to topological quantum field theory and suggesting avenues toward new graph and manifold invariants. The constructions yield computable invariants and open pathways to relate colorings to harmonic/color-based decompositions and potential spin-network–style interpretations in topology.

Abstract

In this paper we show how to categorify the -color vertex polynomial, which is based upon one of Roger Penrose's formulas for counting the number of -edge colorings of a planar trivalent graph. Using topological quantum field theory (TQFT), we introduce a quantum state system to build a new bigraded theory called the bigraded -color vertex homology. The graded Euler characteristic of this homology is the -color vertex polynomial. We then produce a spectral sequence whose -page is a filtered theory called filtered -color vertex homology and show that it is generated by certain types of face colorings of ribbon graphs. For , we show that the filtered -color vertex homology is generated by face colorings that correspond to perfect matchings. Finally, we introduce and give meaning to what the vertex polynomial counts when . This polynomial is a new abstract graph invariant that can be inferred from certain formulas of Penrose.
Paper Structure (30 sections, 37 theorems, 85 equations, 43 figures, 2 tables)

This paper contains 30 sections, 37 theorems, 85 equations, 43 figures, 2 tables.

Table of Contents

  1. introduction
  2. Organization of the paper
  3. Perfect Matching Graphs
  4. Ribbon graphs
  5. Perfect matching graphs
  6. The blowup of a graph
  7. The $n$-Color Vertex Polynomial of a Ribbon Graph
  8. Bigraded $n$-Color Homology
  9. Smoothings, states, and hypercubes
  10. The differential chain complex for $\Gamma$
  11. Vertex Homology
  12. The hypercube of vertex states
  13. The differential chain complex for $\Gamma$
  14. Examples
  15. Filtered $n$-color vertex homology
  16. ...and 15 more sections

Key Result

Theorem 1

Let $G(V,E)$ be a trivalent graph and $\Gamma$ be a ribbon graph of it. Let $n\in {\mathbb N}$ and $\mathbbm{k}$ be a ring in which $\sqrt{n}$ is defined. The bigraded $n$-color vertex homology of $\Gamma$, $VCH_n^{*,*}(\Gamma;\mathbbm{k})$ is an invariant of the ribbon graph $\Gamma$. Furthermore,

Figures (43)

  • Figure 1: Distinct ribbon graphs $\Gamma_1$ and $\Gamma_2$ with the same underlying graph $G$.
  • Figure 2: A ribbon diagram of a $K_{3,3}$ ribbon graph. Note that the closed oriented surface associated to the ribbon graph is a torus.
  • Figure 3: Perfect matching graphs.
  • Figure 4: The bubbled blowup of the theta graph.
  • Figure 5: The blowup and vertex ribbon diagram of the theta graph.
  • ...and 38 more figures

Theorems & Definitions (102)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 2.1
  • Definition 2.2: Ribbon diagram, cf. BKR
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 92 more