Jones-Wenzl projectors and odd Khovanov homology

TL;DR

This paper generalizes categorification techniques for Jones-Wenzl projectors within odd Khovanov homology, establishing their existence and uniqueness, and introduces an 'odd' categorification of the colored Jones polynomial.

Contribution

It extends the categorification framework using grading multicategories, proving the existence and uniqueness of odd Khovanov projectors, and develops a new odd categorification of the colored Jones polynomial.

Findings

Existence and uniqueness of categorified Jones-Wenzl projectors in odd Khovanov homology.

Introduction of a new 'odd' categorification of the colored Jones polynomial.

Generalization of previous categorification approaches using grading multicategories.

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.