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A spanning tree model for Khovanov homology, Rasmussen's s-invariant and exotic discs in the $4$-ball

Aninda Banerjee, Apratim Chakraborty, Swarup Kumar Das

TL;DR

This work provides a fully explicit combinatorial differential for the spanning-tree model of Khovanov homology by encoding chain data in the Tait graph via acyclic matchings and Algebraic discrete Morse theory. It derives a Morse complex whose homology recovers the (reduced and unreduced) Khovanov homology and extends the construction to Lee cohomology, enabling a graph-based description of Rasmussen's s-invariant. A key contribution is the expression of the differential in terms of alternating paths on the spanning-tree Hasse graphs, plus a notion of orientation-preserving trees that tie the s-invariant to the spanning-tree framework. The paper also uses these tools to produce an infinite family of knots that bound exotic ribbon disks in the 4-ball, illustrating a concrete application of the combinatorial model to 4-dimensional topology. Overall, the results integrate Khovanov (and Lee) homology, discrete Morse theory, and Tait-graph combinatorics to yield new computable invariants and topological obstructions with potential extensions to other bi-graded theories.

Abstract

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.

A spanning tree model for Khovanov homology, Rasmussen's s-invariant and exotic discs in the $4$-ball

TL;DR

This work provides a fully explicit combinatorial differential for the spanning-tree model of Khovanov homology by encoding chain data in the Tait graph via acyclic matchings and Algebraic discrete Morse theory. It derives a Morse complex whose homology recovers the (reduced and unreduced) Khovanov homology and extends the construction to Lee cohomology, enabling a graph-based description of Rasmussen's s-invariant. A key contribution is the expression of the differential in terms of alternating paths on the spanning-tree Hasse graphs, plus a notion of orientation-preserving trees that tie the s-invariant to the spanning-tree framework. The paper also uses these tools to produce an infinite family of knots that bound exotic ribbon disks in the 4-ball, illustrating a concrete application of the combinatorial model to 4-dimensional topology. Overall, the results integrate Khovanov (and Lee) homology, discrete Morse theory, and Tait-graph combinatorics to yield new computable invariants and topological obstructions with potential extensions to other bi-graded theories.

Abstract

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 -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.

Paper Structure

This paper contains 22 sections, 40 theorems, 60 equations, 24 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algebraic discrete Morse Theory
  4. Links and Tait graph
  5. Spanning tree and Skein resolution tree
  6. Khovanov homology
  7. Lee cohomology and s-invariant
  8. Reduced spanning tree complex
  9. Acyclic matching on reduced Khovanov complex
  10. Reduced spanning tree complex
  11. Alternating paths between critical enhanced subgraphs
  12. Unreduced spanning tree complex
  13. Acyclic matching on the unreduced Khovanov complex
  14. Spanning tree complex
  15. Torsion in alternating links
  16. ...and 7 more sections

Key Result

Theorem 1.1

For a connected link diagram $\mathcal{L}$, the cohomology of the complex $(\mathop{\mathrm{CST}}\nolimits^+(G_{\mathcal{L}}), \partial_{ST}^+)$ is isomorphic to the reduced Khovanov homology ${\mathop{\mathrm{Kh}}\nolimits}^+(\mathcal{L})$ (Also denoted by $\widetilde{\mathop{\mathrm{Kh}}\nolimits}

Figures (24)

  • Figure 1: The slice disks $D_1$ and $D_2$ bounded by the knot $K$ forms an exotic pair detected by the distinguished oriented resolution cycle under Khovanov cobordism map
  • Figure 2: Right-handed trefoil diagram $D$ and its Tait graph $G_D$
  • Figure 3: Sign of an edge of the Tait graph based on the crossing
  • Figure 4: Skein resolution tree of the left-handed trefoil
  • Figure 6: (a) $e_1$ is a negative twist (b) $e_1$ is a positive twist
  • ...and 19 more figures

Theorems & Definitions (96)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 86 more