A spanning tree model for chromatic homology
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910900933361664 |
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| author | Banerjee, Aninda Chakraborty, Apratim Das, Swarup Kumar Paul, Pravakar |
| author_facet | Banerjee, Aninda Chakraborty, Apratim Das, Swarup Kumar Paul, Pravakar |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_00834 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A spanning tree model for chromatic homology Banerjee, Aninda Chakraborty, Apratim Das, Swarup Kumar Paul, Pravakar Combinatorics Quantum Algebra 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. |
| title | A spanning tree model for chromatic homology |
| topic | Combinatorics Quantum Algebra |
| url | https://arxiv.org/abs/2504.00834 |