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



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)

| cite | improve this answer | |
  • $\begingroup$ Maybe I will add a brief summary of that paper later $\endgroup$ – Siqi He Apr 29 '15 at 17:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.