#### Volume 20, issue 6 (2016)

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

### David E V Rose and Paul Wedrich

Geometry & Topology 20 (2016) 3431–3517
##### 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.