On the quantum filtration of the Khovanov–Rozansky cohomology

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 standard contact S 3 .