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The Functoriality of Odd Khovanov Homology up to Sign and Applications

Jacob Migdail

TL;DR

This work proves that odd Khovanov homology extends to a functor from the link-cobordism category to a chronology-sensitive target category, with functoriality guaranteed up to an overall sign. It develops a dotted version of the theory, yielding a module structure over the exterior algebra of the coloring group, and uses this framework to derive a Hecke-algebra action on the odd Khovanov homology of the $n$-cable of a knot when $n$ is even or the framing is even. The results hinge on carefully defined sign assignments, chronology-aware cobordisms, and detailed verification of all movie moves, Reidemeister moves, and chronological rearrangements. The combination of functoriality up to sign, dotted enhancements, and Hecke-action provides a robust platform for applications to knot concordance and 2-knot invariants, while also clarifying the interplay between odd and even categorifications. Overall, the dissertation broadens the structure and computability of odd Khovanov homology with new algebraic actions and cobordism-functorial perspectives.

Abstract

In this dissertation, we extend the odd Khovanov bracket to link cobordisms and prove that our construction is functorial up to sign. We then build an odd Khovanov theory for dotted link cobordisms. Out of the dotted theory, a module structure on the odd Khovanov homology of a diagram over the exterior algebra of the diagram's coloring group arises. We finish by using our functoriality result to prove that if $n$ is even or if the knot has even framing, then the odd Khovanov homology of the $n$-cable of a knot admits an action of the Hecke algebra $\mathcal{H}(q^2,n)$ at $q=i$.

The Functoriality of Odd Khovanov Homology up to Sign and Applications

TL;DR

This work proves that odd Khovanov homology extends to a functor from the link-cobordism category to a chronology-sensitive target category, with functoriality guaranteed up to an overall sign. It develops a dotted version of the theory, yielding a module structure over the exterior algebra of the coloring group, and uses this framework to derive a Hecke-algebra action on the odd Khovanov homology of the -cable of a knot when is even or the framing is even. The results hinge on carefully defined sign assignments, chronology-aware cobordisms, and detailed verification of all movie moves, Reidemeister moves, and chronological rearrangements. The combination of functoriality up to sign, dotted enhancements, and Hecke-action provides a robust platform for applications to knot concordance and 2-knot invariants, while also clarifying the interplay between odd and even categorifications. Overall, the dissertation broadens the structure and computability of odd Khovanov homology with new algebraic actions and cobordism-functorial perspectives.

Abstract

In this dissertation, we extend the odd Khovanov bracket to link cobordisms and prove that our construction is functorial up to sign. We then build an odd Khovanov theory for dotted link cobordisms. Out of the dotted theory, a module structure on the odd Khovanov homology of a diagram over the exterior algebra of the diagram's coloring group arises. We finish by using our functoriality result to prove that if is even or if the knot has even framing, then the odd Khovanov homology of the -cable of a knot admits an action of the Hecke algebra at .
Paper Structure (86 sections, 25 theorems, 49 equations, 35 figures)

This paper contains 86 sections, 25 theorems, 49 equations, 35 figures.

Table of Contents

  1. Introduction
  2. Cobordisms
  3. Planar Cobordisms
  4. Movies
  5. Elementary Planar Cobordisms
  6. Link Cobordisms
  7. Elementary Link Cobordisms
  8. The Source Category $\mathcal{C}ob^4$
  9. Movie Equivalence in the Source Category
  10. The Target Category
  11. Oriented Planar Cobordisms
  12. Chronological Cobordisms
  13. The Odd Khovanov Homology Planar Cobordism Category
  14. The Final Target Category $\textnormal{Kom}(\textnormal{Mat}(\mathcal{C}hron\mathcal{C}ob^3_{Odd}))$
  15. Odd Khovanov Homology
  16. ...and 71 more sections

Key Result

Theorem 1

Two movies represent smoothly ambient isotopic link cobordisms if and only if they differ by a finite sequence of chronological movie moves, planar ambient isotopy cobordismsIn CS1997 planar ambient isotopy cobordisms are implicit, we explicitly state these as we must also consider if our functor re

Figures (35)

  • Figure 1: On the left is a cobordism with a distinguished frame. On the right is you---the reader---looking at the cobordism on the left, now presented as a movie featuring the same distinguished frame
  • Figure 2: A birth cobordism
  • Figure 3: A death cobordism
  • Figure 4: A merge cobordism
  • Figure 5: A split cobordism
  • ...and 30 more figures

Theorems & Definitions (82)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Definition 10
  • ...and 72 more