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Functoriality of Odd and Generalized Khovanov Homology in $\mathbb{R}^3\times I$

Jacob Migdail, Stephan Wehrli

TL;DR

The paper develops a generalized Khovanov bracket that extends to smooth link cobordisms in $\mathbb{R}^3\times I$ and proves functoriality up to invertible scalars, with a specialization at $\pi=-1$ yielding odd Khovanov functoriality up to sign. It introduces a chronological cobordism framework, establishes two sign-configuration families (Type X and Type Y) and proves their natural equivalence, and constructs entire cobordism maps for births, deaths, saddles, and Reidemeister-type moves, all while maintaining a refined supergrading. The main contributions are the functoriality theorem for the generalized Khovanov bracket, a corrected equivalence result for Type X and Type Y sign choices, and explicit demonstrations that odd Khovanov homology is not functorial under smooth cobordisms in $S^3\times I$, even though it is in $\mathbb{R}^3\times I$. The results illuminate the structural differences between even and odd theories, enable potential 4-manifold invariants via odd Khovanov maps, and set up a robust diagrammatic and TQFT-based framework for further study of cobordism-induced maps.

Abstract

We extend the generalized Khovanov bracket to smooth link cobordisms in $\mathbb{R}^3\times I$ and prove that the resulting theory is functorial up to global invertible scalars. The generalized Khovanov bracket can be specialized to both even and odd Khovanov homology. Particularly by setting $π=-1$, we obtain that odd Khovanov homology is functorial up to sign. We end by showing that odd Khovanov homology is not functorial under smooth link cobordisms in $S^3\times I$.

Functoriality of Odd and Generalized Khovanov Homology in $\mathbb{R}^3\times I$

TL;DR

The paper develops a generalized Khovanov bracket that extends to smooth link cobordisms in and proves functoriality up to invertible scalars, with a specialization at yielding odd Khovanov functoriality up to sign. It introduces a chronological cobordism framework, establishes two sign-configuration families (Type X and Type Y) and proves their natural equivalence, and constructs entire cobordism maps for births, deaths, saddles, and Reidemeister-type moves, all while maintaining a refined supergrading. The main contributions are the functoriality theorem for the generalized Khovanov bracket, a corrected equivalence result for Type X and Type Y sign choices, and explicit demonstrations that odd Khovanov homology is not functorial under smooth cobordisms in , even though it is in . The results illuminate the structural differences between even and odd theories, enable potential 4-manifold invariants via odd Khovanov maps, and set up a robust diagrammatic and TQFT-based framework for further study of cobordism-induced maps.

Abstract

We extend the generalized Khovanov bracket to smooth link cobordisms in and prove that the resulting theory is functorial up to global invertible scalars. The generalized Khovanov bracket can be specialized to both even and odd Khovanov homology. Particularly by setting , we obtain that odd Khovanov homology is functorial up to sign. We end by showing that odd Khovanov homology is not functorial under smooth link cobordisms in .

Paper Structure

This paper contains 39 sections, 21 theorems, 45 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Organization
  3. Preliminaries
  4. The Source Category: $\mathcal{C}ob^4$
  5. The Target Category: $\mathcal{C}hron\mathcal{C}ob^3_{gen}$
  6. Emergent Relations
  7. The Target Category: $\textnormal{Kom}^b(\textnormal{Mat}(\mathcal{C}hron\mathcal{C}ob^3_{gen}))$
  8. Type X and Type Y Configurations
  9. Generalized Khovanov Theory
  10. Crossing Resolution
  11. The Generalized Khovanov Bracket
  12. Supergraded Refinement
  13. Odd and Generalized Khovanov TQFTs
  14. Well Definedness of the Generalized Khovanov Bracket
  15. Dependence on Sign Assignments and Crossing Orientations
  16. ...and 24 more sections

Key Result

Theorem 1

The generalized Khovanov bracket is functorial under smooth link cobordims up to homotopy and overall invertible scalars.

Figures (18)

  • Figure 1: Cobordisms with only one critical point
  • Figure 2: Default orientations used in pictures
  • Figure 3: Cobordisms of type X and type Y configurations
  • Figure 4: The two possible orientations on a crossing
  • Figure 5: The two resolutions of a crossing
  • ...and 13 more figures

Theorems & Definitions (47)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Conjecture 1
  • Theorem 4: CS1993
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • ...and 37 more