# A basic question of Khovanov-Rozansky Homology

thank you for spending time on the following question.

In [1] Khovanov and Rozansky categorifized $sl_n$ version of HOMFLY Polynomial, in page 11, they mention that what they defined in [1] is equivalence to what Khovanov "original" defined in [2] and give a short argument:

(I) For a closed link diagram $\Gamma$ with marked point, its cohomology group $H(\Gamma)$ are isomorphic to $A_2^{\otimes k}$, where $A_2$ is the cohomology ring of 2-sphere and k the number of circles in the diagram given by deleting all wide edges of $\Gamma$

(II)The equivalence of n=2 and [2] follows easily from observation.

I wonder that anyone could tell me how to show the equivalence in detail or give me some hint for proving (I) and (II). Even in n=2, I still find unable to understand the factorization for a wide edge with potential $w_t=x_1^{n+1}+x_2^{n+1}-x_3^{n+1}-x_4^{n+1}$.

[1]Matrix Factorizations and Link Homology, Khovanov, Rozansky

[2]A Categorification of the Jones Polynomial, Khovanov