TITLE: HOMFLY and homology; also superalgebras
QUESTION [16 upvotes]: My understanding is that an analogy along the following lines is (roughly) true: 
"The Alexander polynomial is to knot Floer homology is to gl(1|1)
as the Jones polynomial is to Khovanov homology is to sl(2)
as a-lot-of-other-specializations-of-HOMFLY are to Khovanov-Rozansky homology are to sl(n)."
1) To what extent is it possible to add another line that starts with the (unspecialized) HOMFLY polynomial?  I think there is a triply-graded complex that I can put here (and that maybe this is what I should be calling Khovanov-Rozansky homology? or at least is also due to them?), but is there an analogous object to put in place of the Lie (super-)algebras appearing above?
2) Why is gl(1|1) here? That seems weird.

REPLY [10 votes]: Lots of good answers above, but the one that seems to be missing is what belongs in the third column for unspecialized HOMFLY.  (As Geordie Williamson points out, Khovanov and Rozansky wrote two related papers, one of which does the unspecialized case.)  There's a one parameter family of categories that looks like the category of representations of a Lie superalgebra, called $gl(\alpha)$.  When $\alpha$ is an integer, you can kill off the representations that have dimension 0 and get the usual category of representations of $gl(n)$.  (You can also kill off fewer representations and get the representations of $gl(n+k|k)$ for any $k$.)  Many people have written about this, but perhaps the coolest place to look is in Deligne's La série exceptionelle de groupes de Lie, C. R. Acad. Sci. Paris, Série I 322 (1996) 321—326, where he also conjectures a similar structure specializing to all the exceptional groups.  (There are also several follow-up papers.)