Generic deformations of the colored sl(N)-homology for links

TL;DR

This paper develops a framework for generic deformations of colored sl(N)-homology, introduces dualities and invariants, and explores their properties, including vanishing on amphicheiral knots, advancing link homology theory.

Contribution

It constructs a basis for generic deformations of colored sl(N)-homology and introduces dualities and Rasmussen invariants, extending previous work in link homology.

Findings

Constructed a basis for generic deformations of colored sl(N)-homology.

Established dualities via non-degenerate pairings and co-pairings.

Defined colored sl(N)-Rasmussen invariants and observed their vanishing on amphicheiral knots.

Abstract

We generalize the works of Lee [arXiv:math/0210213v3] and Gornik [arXiv:math/0402266v2] to construct a basis for generic deformations of the colored sl(N)-homology defined in [arXiv:1002.2662v1]. As applications, we construct non-degenerate pairings and co-pairings which lead to dualities of generic deformations of the colored sl(N)-homology. We also define and study colored sl(N)-Rasmussen invariants. Among other things, we observe that these invariants vanish on amphicheiral knots and discuss some implications of this observation.