Matrix factorizations and link homology
Mikhail Khovanov, Lev Rozansky
TL;DR
The paper constructs a family of doubly-graded link homologies $H_n(L)$ for $n>0$ whose Euler characteristics recover the HOMFLY polynomial $P_n(L)$ by employing matrix factorizations tied to planar MOY graphs. It develops a robust, diagrammatic calculus based on cyclic Koszul complexes and graded factorizations, enabling local graph decompositions and a tangle-based, functorial invariant framework. The approach yields a combinatorial algorithm to compute $H_n(L)$ and connects to established theories for $n=0,1,2,3$, while outlining conjectural correspondences for $n=3$ and potential extensions to tangle cobordisms. Overall, the work provides a unified algebraic pathway to categorify quantum-group–related link invariants via matrix factorizations and planar graph calculus, with broad implications for link homology theories and topological quantum field theories.
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.