Affine Braid group, JM elements and knot homology

Abstract

In this paper we construct a homomorphism of the affine braid group $B{r}_n^{aff}$ in the convolution algebra of the equivariant matrix factorizations on the space ${\overline{\mathcal{X}}}_2={\mathfrak{b}}_n\times G{L}_n\times {\mathfrak{n}}_n$ considered in the earlier paper of the authors. We explain that the pull-back on the stable part of the space $\overline{{\mathcal{X}}_2}$ intertwines with the natural homomorphism from the affine braid group $B{r}_n^{aff}$ to the finite braid group $B{r}_n$. This observation allows us derive a relation between the knot homology of the closure of $\beta \in B{r}_n$ and the knot homology of the closure of $\beta \cdot \delta$ where $\delta$ is a product of the JM elements in $B{r}_n$

Many typos are corrected, published version, to appear in Transformation Groups

References in corpus (3)

• Compact generators in categories of matrix factorizations
• The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link
• Commuting Families in Temperley-Lieb Algebras