Computing colored Khovanov homology

TL;DR

This paper compares eight categorifications of the colored Jones polynomial, demonstrating their isomorphism over characteristic zero fields, and verifies a conjectural formula for colored superpolynomials with additional related results.

Contribution

It establishes the equivalence of multiple categorifications of the colored Jones polynomial and verifies a physics-inspired conjecture for colored superpolynomials.

Findings

Eight categorifications are isomorphic over characteristic zero.

Verification of a conjectural formula for colored superpolynomials.

Provides an online database of colored superpolynomials.

Abstract

We compare eight versions of finite-dimensional categorifications of the colored Jones polynomial and show that they yield isomorphic results over a field of characteristic zero. As an application, we verify a physics-motivated conjectural formula for colored superpolynomials based on Poincar\'e polynomials of the Khovanov homology of cables. We also obtain a conjectural closed formula for the Poincar\'e series of the skein lasagna module of $\overline{\mathbb{CP}^2}$. Accompanying this note is an online database of colored superpolynomials.