Categorification of the Jones-Wenzl Projectors

TL;DR

This paper develops a categorification of Jones-Wenzl projectors using chain complexes, leading to new knot invariants and a deeper understanding of quantum spin networks within the framework of Khovanov's categorification.

Contribution

It constructs homotopy idempotent chain complexes representing Jones-Wenzl projectors, extending Khovanov's categorification to quantum spin networks and higher representations.

Findings

Constructed chain complexes with Euler characteristic equal to classical projectors

Proved these complexes are homotopy idempotents and unique up to homotopy

Introduced categorified 6j-symbols and new knot invariants

Abstract

The Jones-Wenzl projectors play a central role in quantum topology, underlying the construction of SU(2) topological quantum field theories and quantum spin networks. We construct chain complexes whose graded Euler characteristic is the "classical" projector in the Temperley-Lieb algebra. We show that they are homotopy idempotents and uniquely defined up to homotopy. Our results fit within the general framework of Khovanov's categorification of the Jones polynomial. Consequences of our construction include families of knot invariants corresponding to higher representations of quantum su(2) and a categorification of quantum spin networks. We introduce 6j-symbols in this context.