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.