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Jones-Wenzl projectors and odd Khovanov homology

Dean Spyropoulos

Abstract

The Jones-Wenzl projectors are particular elements of the Temperley-Lieb algebra essential to the construction of quantum 3-manifold invariants. As a first step toward categorifying quantum 3-manifold invariants, Cooper and Krushkal categorified these projectors. In another direction, Naisse and Putyra gave a categorification of the Temperley-Lieb algebra compatible with odd Khovanov homology, introducing new machinery called grading categories. We provide a generalization of Naisse and Putyra's work in the spirit of Bar-Natan's canopolies or Jones's planar algebras, replacing grading categories with grading multicategories. We use our setup to prove the existence and uniqueness of categorified Jones-Wenzl projectors in odd Khovanov homology. This result quickly implies the existence of a new, "odd" categorification of the colored Jones polynomial.

Jones-Wenzl projectors and odd Khovanov homology

Abstract

The Jones-Wenzl projectors are particular elements of the Temperley-Lieb algebra essential to the construction of quantum 3-manifold invariants. As a first step toward categorifying quantum 3-manifold invariants, Cooper and Krushkal categorified these projectors. In another direction, Naisse and Putyra gave a categorification of the Temperley-Lieb algebra compatible with odd Khovanov homology, introducing new machinery called grading categories. We provide a generalization of Naisse and Putyra's work in the spirit of Bar-Natan's canopolies or Jones's planar algebras, replacing grading categories with grading multicategories. We use our setup to prove the existence and uniqueness of categorified Jones-Wenzl projectors in odd Khovanov homology. This result quickly implies the existence of a new, "odd" categorification of the colored Jones polynomial.

Paper Structure

This paper contains 52 sections, 69 theorems, 454 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Unified projectors
  3. A flexible gluing property
  4. Other applications
  5. Future goals
  6. Categorifications of Lg and projectors
  7. Temperley-Lieb algebras and Jones-Wenzl projectors
  8. Categorifications of the Temperley-Lieb algebra
  9. Chain homotopy lemmas
  10. Khovanov's arc algebras
  11. Cooper-Krushkal projectors
  12. The Odd setting: chronologies and Lg-graded structures
  13. Chronological cobordisms and changes of chronology
  14. Unified arc algebras
  15. A brief outline of Lg-graded structures
  16. ...and 37 more sections

Key Result

Theorem 1

There exist categorifications of the Jones-Wenzl projectors, called unified projectors, $P_n$ in $\mathrm{Chom}(n)_R^\mathscr{G}$, which are unique up to chain-homotopy equivalence. By a categorification, we mean that $[P_n] \in K_0^q(\mathrm{Chom}(n)_R^\mathscr{G})$ is equal to $p_n\in TL_n$ (for a

Figures (6)

  • Figure 1: Schematic for $\Pi^\mathbf{m}(L)$, where $L$ is the 3-component link L11n314 of the Thistelthwaite link table, and $\mathbf{m} = (3,2,2)$.
  • Figure 2: Multigluing schematic. Here, we assume $T_1$, $T_2$, and $T_3$ are each tangles in disks with 4 points on their boundary.
  • Figure 3: This is the collection of elementary changes of chronologies, together with their evaluation by $\iota$. Notice that taking $X=Z=1$ and $Y=-1$ yields the commutation chart of Ozsv_th_2013. Framings are omitted if evaluation by $\iota$ does not depend on them.
  • Figure 4: An example of a chronological coboridm $W_{\vec{x}\vec{y}z}((D_1, D_2, D_3), D)$.
  • Figure 5: Computing the grading shift on $\mathcal{T}_4^4 \otimes e_1$.
  • ...and 1 more figures

Theorems & Definitions (157)

  • Theorem : Theorem A
  • Theorem : Theorem B
  • Theorem : Theorem C
  • Theorem : Theorem \ref{['thm:adjunction1']}
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • ...and 147 more