A Lie theoretic categorification of the coloured Jones polynomial

TL;DR

This paper develops a new categorification framework for the coloured Jones polynomial of links using Lie theoretic methods and categorified Jones-Wenzl projectors, extending the invariants to arbitrary finite-dimensional representations.

Contribution

It introduces a Lie theoretic categorification approach for the coloured Jones polynomial, generalizing previous invariants to a broader class of representations.

Findings

Constructed a categorification of type A Reshetikhin-Turaev invariants for framed tangles.

Achieved a categorification of the coloured Jones polynomial for links.

Extended the categorification to arbitrary finite-dimensional representations.

Abstract

We use the machinery of categorified Jones-Wenzl projectors to construct a categorification of a type A Reshetikhin-Turaev invariant of oriented framed tangles where each strand is labeled by an arbitrary finite-dimensional representation. As a special case, we obtain a categorification of the coloured Jones polynomial of links.