On the quantum filtration of the universal sl(2) foam cohomology

TL;DR

This paper explores a filtered version of the universal sl(2) foam cohomology for links, establishing a spectral sequence from Khovanov homology to this cohomology, and deriving applications like a Rasmussen-type invariant and slice genus bounds.

Contribution

It introduces a spectral sequence invariant under Reidemeister moves connecting Khovanov homology to universal sl(2) foam cohomology, enabling new knot invariants.

Findings

Spectral sequence converges to $H_{a,h}$ from Khovanov homology.

Invariant under Reidemeister moves.

Provides a lower bound for the slice genus.

Abstract

We investigate the filtered theory corresponding to the universal sl(2) foam cohomology $H_{a,h}$ for links, where a and h are complex numbers. We show that there is a spectral sequence converging to $H_{a,h}$ which is invariant under the Reidemeister moves, and whose E1 term is isomorphic to Khovanov homology. This spectral sequence can be used to obtain from the foam perspective an analogue of the Rasmussen invariant and a lower bound for the slice genus of a knot.