On the quantum filtration of the Khovanov-Rozansky cohomology

TL;DR

This paper establishes the invariance of the quantum filtration on Khovanov-Rozansky cohomology under Reidemeister moves, constructs a spectral sequence linking it to sl(n)-cohomology, and explores a generalized Rasmussen invariant.

Contribution

It proves the invariance of the quantum filtration for general potentials and introduces a spectral sequence connecting it to known cohomology theories, along with a new invariant.

Findings

Quantum filtration is invariant under Reidemeister moves.

Constructed a spectral sequence converging to H_p with E_1 isomorphic to H_n.

Defined and analyzed a generalized Rasmussen invariant.

Abstract

We prove the quantum filtration on the Khovanov-Rozansky link cohomology H_p with a general degree (n+1) monic potential polynomial p(x) is invariant under Reidemeister moves, and construct a spectral sequence converging to H_p that is invariant under Reidemeister moves, whose E_1 term is isomorphic to the Khovanov-Rozansky sl(n)-cohomology H_n. Then we define a generalization of the Rasmussen invariant, and study some of its properties. We also discuss relations between upper bounds of the self-linking number of transversal links in the standard contact three sphere.