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Odd Khovanov homology and higher representation theory

Léo Schelstraete, Pedro Vaz

Abstract

We define a supercategorification of the $q$-Schur algebra of level two and an odd analogue of $\mathfrak{gl}_2$-foams. Using these constructions, we define a homological invariant of tangles, and show that it coincides with odd Khovanov homology when restricted to links. This gives a representation theoretic construction of odd Khovanov homology. In the process, we define a tensor product on the category of chain complexes in super-2-categories which is compatible with homotopies. This could be of independent interest.

Odd Khovanov homology and higher representation theory

Abstract

We define a supercategorification of the -Schur algebra of level two and an odd analogue of -foams. Using these constructions, we define a homological invariant of tangles, and show that it coincides with odd Khovanov homology when restricted to links. This gives a representation theoretic construction of odd Khovanov homology. In the process, we define a tensor product on the category of chain complexes in super-2-categories which is compatible with homotopies. This could be of independent interest.
Paper Structure (20 sections, 9 theorems, 47 equations, 4 figures)

This paper contains 20 sections, 9 theorems, 47 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Extended summary
  4. Super-2-categories
  5. Odd Khovanov homology
  6. Super gl2-foams
  7. Odd Khovanov homology is a super tensor product of chain complexes
  8. A supercategorification of the q-Schur algebra of level two
  9. A Graded-2-category of gl2-foams
  10. Graded-2-categories
  11. Diagrammatics for graded-2-categories
  12. More categorical preliminaries
  13. gl2-webs
  14. Graded gl2-foams
  15. String diagrammatics
  16. ...and 5 more sections

Key Result

Lemma 2.10

Let $W$ and $W'$ be two ${\mathfrak{gl}_{2}}$-webs with the same domain and codomain. Then $W$ and $W'$ are equal in $\mathbf{Web}_d$ if and only if there exists a spatial isotopy between $c(W)$ and $c(W')$.

Figures (4)

  • Figure 2.1: Local model for foams.
  • Figure 2.2: Relations in ${\mathbf{GFoam}}_d$. A grey dot denotes a dot behind a facet. The $\mathbb{R}$ axis is pictured from front to back, except for the squeezing relation for which it is pictured from left to right for better readability.
  • Figure 2.3: Relations in ${\mathbf{Diag}}_d^\Lambda$. We omitted the objects label the regions of each diagram: this avoids clutter and emphasizes the relations' independence on objects. If no orientation is given, the relation holds for all orientations. In the case of the braid-like and pitchfork relations, colours should be so that the crossings exist.
  • Figure 3.1:

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7: Orientation
  • Definition 2.8
  • Definition 2.9
  • Lemma 2.10
  • ...and 24 more