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 Poincaré polynomials of the Khovanov homology of cables. We also obtain a conjectural closed formula for the Poincaré series of the skein lasagna module of $\overline{{\mathbb{CP}}^2}$. Accompanying this note is an online database of colored superpolynomials.

17 pages, v3: implementing referee comments and fixing typos, to appear in Algebraic & Geometric Topology