# Notes on matrix factorizations and knot homology

## TL;DR

This paper provides an overview of techniques for constructing triply graded link homology using sheaves on Hilbert schemes, emphasizing matrix factorizations and localization formulas.

## Contribution

It introduces a novel approach to knot homology via equivariant matrix factorizations and details the construction, including proofs of Markov moves and localization methods.

## Key findings

- Construction of triply graded link homology using sheaves on Hilbert schemes
- Explicit localization formulas for knot homology of various links
- Outline of proof for Markov moves in the context of this homology

## Abstract

These are the notes of the lectures delivered by the author at CIME in June 2018. The main purpose of the notes is to provide an overview of the techniques used in the construction of the triply graded link homology. The homology is space of global sections of a particular sheaf on the Hilbert scheme of points on the plane. Our construction relies on existence on the natural push-forward functor for the equivariant matrix factorizations, we explain the subtleties on the construction in these notes. We also outline a proof of the Markov moves for our homology as well as some explicit localization formulas for knot homology of a large class of links.