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.

reallysilly thing to think about, and gl(1|1) steps in and prevents us from having to be so silly. Now a) this is really cool, and b) I can rephrase (part of) my question as "why is gl(1|1) the correct substitute for sl(0)?" $\endgroup$ – Harold Williams Oct 22 '09 at 20:15