Matrix factorizations and link homology

TL;DR

This paper constructs a doubly-graded link homology theory for each positive integer n, connecting HOMFLY polynomials with quantum sl(n) representation theory via matrix factorizations.

Contribution

It introduces a new link homology theory based on matrix factorizations that generalizes existing invariants and relates to quantum group representations.

Findings

Defines a homology theory with the HOMFLY polynomial as Euler characteristic

Establishes a link between matrix factorizations and Cohen-Macaulay modules

Provides a new algebraic framework for link invariants

Abstract

For each positive integer n the HOMFLY polynomial of links specializes to a one-variable polynomial that can be recovered from the representation theory of quantum sl(n). For each such n we build a doubly-graded homology theory of links with this polynomial as the Euler characteristic. The core of our construction utilizes the theory of matrix factorizations, which provide a linear algebra description of maximal Cohen-Macaulay modules on isolated hypersurface singularities.