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



A basic question of Khovanov-Rozansky Homology
<http://www.worldscientific.com/doi/abs/10.1142/S0218216514500576>


In the following paper, the author give a explicit isomorphism.


Mark C. Hughes, A note on Khovanov–Rozansky sl2-homology and ordinary Khovanov homology, J. Knot Theory Ramifications, 23, 1450057 (2014)


