# Affine Braid group, JM elements and knot homology

## TL;DR

This paper constructs a homomorphism from the affine braid group to a convolution algebra of matrix factorizations, linking it to knot homology and revealing relations involving JM elements.

## Contribution

It introduces a new homomorphism connecting affine braid groups with matrix factorizations, elucidating their role in knot homology computations.

## Key findings

- Established a homomorphism from affine braid group to matrix factorizations
- Demonstrated how JM elements relate different knot homologies
- Linked affine braid group actions to knot invariants

## Abstract

In this paper we construct a homomorphism of the affine braid group $Br_n^{aff}$ in the convolution algebra of the equivariant matrix factorizations on the space $\overline{\mathcal{X}}_2=\mathfrak{b}_n\times GL_n\times\mathfrak{n}_n$ 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 $Br_n^{aff}$ to the finite braid group $Br_n$. This observation allows us derive a relation between the knot homology of the closure of $\beta\in Br_n$ and the knot homology of the closure of $\beta\cdot\delta$ where $\delta$ is a product of the JM elements in $Br_n$