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In arXiv:math/0607544 (following conjectures in arXiv:math/0505662), Rasmussen constructs a family of spectral sequences (the "d_N differentials"), starting at the HOMFLY homology of a knot, and converging to the sl_N Khovanov-Rozansky homology of the knot.

My question is: are there expected to be "intermediate" spectral sequences connecting sl_N homology and sl_M homology for various N and M? Less optimistically, is it known whether the total rank of the sl_N homology is increasing in N (in the "unstable" region before it reaches the HOMFLY homology)?

E.g. the d_1 differential, as constructed by Rasmussen, starts at the HOMFLY homology and converges to the Lee homology. In its original construction, though, Lee's spectral sequence started at the standard ("sl_2") Khovanov homology. Is there a way to extract the original construction from the HOMFLY version?

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Yes. I think the existence of these sequences is "general knowledge" amongst knot homologists, though I'll be honest, I'm not sure where they are written down at the moment. Had you asked me 10 minutes ago, I would have said they were in the paper of Rasmussen you cite, but they seem to not actually be written down (though they are secretly there).

The point is just that you always have spectral sequences from the homology for "more special" potentials to "less special" ones. Since 0 is special as things get, you always have a spectral sequence from HOMFLY homology to the homology attached to any potential. To get the spectral sequence from N to M for $N>M$, just consider the potential built from $p(x)=x^N+a_1x^{N-1}+\cdots + a_{N-M}x^M$ (using Rasmussen's notation); if we consider $a_i$ as free variables, we get sl_M homology with some boring junk, and if we set $a_M=0$, we get sl_N homology. Since the latter is a specialization of the former we get a spectral sequence.

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In the paper by Gukov and Stosic, they formulate the axioms which colored HOMFLY homology (triply-graded homology) is supposed to satisfy, assuming there exists such homology. If you apply the $d_M$ differential on colored HOMFLY homology, then you will obtain colored $sl(M)$-homology. The action of $d_M$ differential is trivial for thin knot while they acts non-trivially on thick knot homology. So far, it has been proven that the axioms work consistently in the classes of the (2,2p+1)-torus knots and the twist knots.

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