A spanning tree model for Khovanov homology, Rasmussen's s-invariant and exotic discs in the $4$-ball
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915225874202624 |
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| author | Banerjee, Aninda Chakraborty, Apratim Das, Swarup Kumar |
| author_facet | Banerjee, Aninda Chakraborty, Apratim Das, Swarup Kumar |
| contents | The checkerboard coloring of knot diagrams offers a graph-theoretical approach to address topological questions. Champanerkar and Kofman defined a complex generated by the spanning trees of a graph obtained from the checkerboard coloring whose homology is the reduced Khovanov homology. Notably, the differential in their chain complex was not explicitly defined. We explicitly define the combinatorial form of the differential within the spanning tree complex. We additionally provide a description of Rasmussen's $s$-invariant within the context of the spanning tree complex. Applying our techniques, we identify a new infinite family of knots where each of them bounds a set of exotic discs within the 4-ball. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_02625 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A spanning tree model for Khovanov homology, Rasmussen's s-invariant and exotic discs in the $4$-ball Banerjee, Aninda Chakraborty, Apratim Das, Swarup Kumar Geometric Topology Combinatorics Quantum Algebra The checkerboard coloring of knot diagrams offers a graph-theoretical approach to address topological questions. Champanerkar and Kofman defined a complex generated by the spanning trees of a graph obtained from the checkerboard coloring whose homology is the reduced Khovanov homology. Notably, the differential in their chain complex was not explicitly defined. We explicitly define the combinatorial form of the differential within the spanning tree complex. We additionally provide a description of Rasmussen's $s$-invariant within the context of the spanning tree complex. Applying our techniques, we identify a new infinite family of knots where each of them bounds a set of exotic discs within the 4-ball. |
| title | A spanning tree model for Khovanov homology, Rasmussen's s-invariant and exotic discs in the $4$-ball |
| topic | Geometric Topology Combinatorics Quantum Algebra |
| url | https://arxiv.org/abs/2504.02625 |