Computing colored Khovanov homology

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.