Deformations of colored sl(N) link homologies via foams

TL;DR

This paper extends the understanding of deformed colored sl(N) link homologies to non-generic cases, using foam technology for a combinatorial construction and relating invariants to lower-rank undeformed homologies.

Contribution

It introduces a combinatorial foam-based construction for non-generic deformed colored sl(N) link homologies and relates them explicitly to lower-rank undeformed invariants.

Findings

Provides a combinatorial model for non-generic deformed link homologies.

Explicitly computes deformed invariants in terms of lower-rank undeformed homologies.

Generalizes previous results to broader deformation settings.

Abstract

We generalize results of Lee, Gornik and Wu on the structure of deformed colored sl(N) link homologies to the case of non-generic deformations. To this end, we use foam technology to give a completely combinatorial construction of Wu's deformed colored sl(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.