# Khovanov-Rozansky homology and spectral sequences

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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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.
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.