# Khovanov-Rozansky $sl_2$ homology and the “original” Khovanov homology.

I'm trying to understand the connection between Khovanov's original link homology and the $sl_2$ version of Khovanov-Rozansky homology. They both categorify the same link polynomial, but is there a direct relationship known between the theories on the level of homology?

Even more optimistically, is there any connection known relating the construction of the original Khovanov homology (in terms of enhanced states, for instance) and the matrix factorization construction of KR homology (when $n=2$)?

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## 1 Answer

In Khovanov and Rozansky's paper "Matrix Factorizations and Link Homology", on p. 11 there is a specific isomorphism between matrix factorization homology of a link $L$ when $n=2$ and a (regraded, rationalized) version of Khovanov homology for the mirror image $L^!$ of $L$.

In Jacob Rasmussen's paper "Some Differentials on Khovanov-Rozansky Homology," on p. 24 there are some comments about cube complexes of resolutions in the context of matrix factorization homology.

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