Some differentials on Khovanov-Rozansky homology

TL;DR

This paper explores the connection between HOMFLY and sl(N) knot homologies, establishing a spectral sequence linking them, and computes KR-homology for knots with up to 9 crossings.

Contribution

It introduces a spectral sequence from HOMFLY to sl(N) homology and computes KR-homology for small knots, advancing understanding of knot homology relationships.

Findings

Spectral sequence from HOMFLY to sl(N) homology established

KR-homology computed for knots with 9 or fewer crossings

Enhanced understanding of knot homology structures

Abstract

We study the relationship between the HOMFLY and sl(N) knot homologies introduced by Khovanov and Rozansky. For each N>0, we show there is a spectral sequence which starts at the HOMFLY homology and converges to the sl(N) homology. As an application, we determine the KR-homology of knots with 9 crossings or fewer.