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Matrix factorizations and link homology
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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.
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Reductions in Khovanov-Rozansky operator formalism
Khovanov-Rozansky invariants are recast as a bicomplex of local operators D and conjugations χ^(±), with nilpotency on closed diagrams allowing reductions that simplify the hypercube construction.
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