Deformations of colored 𝔰𝔩N link homologies via foams

Abstract

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.

Functoriality of colored link homologies

“…In the uncolored case, these agree with the original Khovanov–Rozansky link homologies . More generally, the alphabet can be specialized to any ‐element multiset of complex numbers, and one obtains deformed Khovanov–Rozansky link homologies , see . In the next section, we will explain this in detail for .…”

“…Proof These relations follow immediately from [, Relations (3.13) and (3.14)] as well as [, Relation (4–7)], imported via Proposition , and .…”

“…The canopolis can be seen as its generic deformation. Deformations obtained as other specializations of the variables in have been studied in .…”

“…Proof The left isomorphism is a special case of [, Lemma 5.9], the right isomorphism follows by applying to .…”

“…Proof. See[47, Lemma 4.2] and also[47, proof of Theorem 3.11].Lemma 2.29 [47, Lemma 4.2]. Let A, B and X be subsets of Σ with |A| = a, |B| = a and |X| = a + b respectively.…”