AFFINE BRAID GROUP, JM ELEMENTS AND KNOT HOMOLOGY

In this paper we construct a homomorphism of the affine braid group \( {\mathfrak{Br}}_n^{\mathrm{aff}} \) in the convolution algebra of the equivariant matrix factorizations on the space \( {\overline{\mathcal{X}}}_2={\mathfrak{b}}_n\times {\mathrm{GL}}_n\times {\mathfrak{n}}_n \) 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 \( {\mathfrak{Br}}_n^{\mathrm{aff}} \) to the finite braid group \( {\mathfrak{Br}}_n \). This observation allows us derive a relation between the knot homology of the closure of β ∈ \( {\mathfrak{Br}}_n \) and the knot homology of the closure of β · δ where δ is a product of the JM elements in \( {\mathfrak{Br}}_n \)