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?



Khovanov-Rozansky homology and spectral sequences
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.


