Deformations of colored $\mathfrak{sl}_{N}$ link homologies via foams

Abstract

We prove a conjectured decomposition of deformed ${\mathfrak{s}\mathfrak{l}}_N$ link homology, as well as an extension to the case of colored links, generalizing results of Lee, Gornik, and Wu. To this end, we use foam technology to give a completely combinatorial construction of Wu’s deformed colored ${\mathfrak{s}\mathfrak{l}}_N$ link homologies. By studying the underlying deformed higher representation-theoretic structures and generalizing the Karoubi envelope approach of Bar-Natan and Morrison, we explicitly compute the deformed invariants in terms of undeformed type A link homologies of lower rank and color.