# HOMFLY and homology; also superalgebras

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.

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Maybe just that gl(1|1) is the smallest algebra other than sl(2)? Noah once told me that these categorical constructions are very bag at telling apart honest groups and supergroups. –  Theo Johnson-Freyd Oct 22 '09 at 16:14
sl(1) is smaller than sl(2) :) That being said you're correct in the following sense: there is really a correspondence here between &#8484; and a bunch of algebras, where n>0 ~ sl(n), n<0 ~ sl(-n), and n=0 ~ gl(1|1). So even though sl(1) is kind of a silly think to think about, sl(0) is a really silly 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)?" –  Harold Williams Oct 22 '09 at 20:15

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