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A spanning tree model for chromatic homology

Aninda Banerjee, Apratim Chakraborty, Swarup Kumar Das, Pravakar Paul

TL;DR

The paper presents a spanning tree model for chromatic homology by constructing a Spanning Tree Complex that is chain homotopy equivalent to the original chromatic complex, enabling a combinatorial, tree-based computation of chromatic homology. Using algebraic Morse theory, it derives an explicit differential and proves a spanning-tree expansion for the chromatic polynomial, linking NBC spanning trees to homological data. It extends these ideas to the family of algebras ${\mathcal A}_m$, establishing the homological span $hspan(H^{\mathcal{A}_m}_{*,*}(G))=v-b$ and proving the existence of torsion of order dividing $m$ under stated graph conditions. These results provide a concrete categorification framework and a practical tool for computing chromatic homology via NBC trees, with implications for graph polynomials and their torsion phenomena.

Abstract

After the discovery of Khovanov homology, which categorifies the Jones polynomial, an analogous categorification of the chromatic polynomial, known as chromatic homology, was introduced. Its graded Euler characteristic recovers the chromatic polynomial. In this paper, we present a spanning tree model for the chromatic complex, i.e., we describe a chain complex generated by certain spanning trees of the graph that is chain homotopy equivalent to the chromatic complex. We employ the spanning tree model over $\mathcal{A}_m:= \frac{\mathbb{Z}[x]}{<x^m>}$ algebra to answer two open questions. First, we establish the conjecture posed by Sazdanovic and Scofield regarding the homological span of chromatic homology over $\\mathcal{A}_m$ algebra, demonstrating that for any graph $G$ with $v$ vertices and $b$ blocks, the homological span is $v - b$. Additionally, we prove a conjecture of Helme-Guizon, Przytycki, and Rong concerning the existence of torsion of order dividing $m$ in chromatic homology over $\mathcal{A}_m$ algebra.

A spanning tree model for chromatic homology

TL;DR

The paper presents a spanning tree model for chromatic homology by constructing a Spanning Tree Complex that is chain homotopy equivalent to the original chromatic complex, enabling a combinatorial, tree-based computation of chromatic homology. Using algebraic Morse theory, it derives an explicit differential and proves a spanning-tree expansion for the chromatic polynomial, linking NBC spanning trees to homological data. It extends these ideas to the family of algebras , establishing the homological span and proving the existence of torsion of order dividing under stated graph conditions. These results provide a concrete categorification framework and a practical tool for computing chromatic homology via NBC trees, with implications for graph polynomials and their torsion phenomena.

Abstract

After the discovery of Khovanov homology, which categorifies the Jones polynomial, an analogous categorification of the chromatic polynomial, known as chromatic homology, was introduced. Its graded Euler characteristic recovers the chromatic polynomial. In this paper, we present a spanning tree model for the chromatic complex, i.e., we describe a chain complex generated by certain spanning trees of the graph that is chain homotopy equivalent to the chromatic complex. We employ the spanning tree model over algebra to answer two open questions. First, we establish the conjecture posed by Sazdanovic and Scofield regarding the homological span of chromatic homology over algebra, demonstrating that for any graph with vertices and blocks, the homological span is . Additionally, we prove a conjecture of Helme-Guizon, Przytycki, and Rong concerning the existence of torsion of order dividing in chromatic homology over algebra.

Paper Structure

This paper contains 18 sections, 32 theorems, 98 equations, 18 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Algebraic Morse Theory
  4. Shellable Complex and Posets
  5. A Short Review of Graph Co-homology
  6. Chain Groups
  7. Differential
  8. Broken circuit Complex
  9. Activity word on a spanning tree
  10. Spanning Tree Model for Chromatic Homology
  11. Acyclic matching on chromatic complex of a tree
  12. Acyclic matching on the chromatic complex of a graph G
  13. Spanning Tree Complex
  14. Combinatorial form of the differential of the spanning tree complex
  15. Computations
  16. ...and 3 more sections

Key Result

Theorem 1.1

The spanning tree complex ${CST_{*,*}(G)}$ and the chromatic complex ${C_{*,*}(G)}$ are chain homotopic and therefore, ${HST_{i,j}(G)} \cong {H_{i,j}(G)}$.

Figures (18)

  • Figure 1: A graph G and the Hasse diagram of the face poset $P(2^E)$
  • Figure 2: Chromatic chain groups in terms of enhanced spanning subgraphs for the graph $P_3$.
  • Figure 3: Differential between the chain groups $C_0(P_3)$ and $C_1(P_3)$
  • Figure 4: Comparison of graph $G$ and tree $T$
  • Figure 5: Schematic picture
  • ...and 13 more figures

Theorems & Definitions (79)

  • Theorem 1.1
  • Corollary 1.2
  • Conjecture 1.3: Conjecture 43
  • Conjecture 1.4: Conjecture 25
  • Definition 2.1: Morse Complex
  • Lemma 2.2
  • Theorem 2.3: article
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • ...and 69 more