Some differentials on Khovanov–Rozansky homology

Abstract

We study the relationship between the HOMFLY and $\mathit{sl}\left(N\right)$ 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 $\mathit{sl}\left(N\right)$ homology. As an application, we determine the KR–homology of knots with 9 crossings or fewer.