Generic deformations of the colored $\mathfrak{sl}(N)$–homology for links

Abstract

We generalize the works of Lee [Adv. Math. 197 (2005) 554–586] and Gornik [arXiv math.QA/0402266] to construct a basis for generic deformations of the colored $\mathfrak{s}\mathfrak{l}\left(N\right)$–homology defined in [arXiv 1002.2662v2]. As applications, we construct nondegenerate pairings and co-pairings which lead to dualities of generic deformations of the colored $\mathfrak{s}\mathfrak{l}\left(N\right)$–homology. We also define and study colored $\mathfrak{s}\mathfrak{l}\left(N\right)$–Rasmussen invariants. Among other things, we observe that these invariants vanish on amphicheiral knots and discuss some implications of this observation.