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.
Best Answer
In terms of just the knot polynomials, there's a simple explanation for what's going on that makes $\mathfrak{gl}(1|1)$ seem totally in place:
The knot polynomial attached to the defining representation of $\mathfrak{gl}(m|n)$ only depends on m-n (the dimension of that representation in the categorical sense); you just get the specialization of HOMFLY at $t=q^{m-n}$.
Furthermore, nothing much interesting happens at negative values, since they're basically the same as positive ones. So, our current techniques, which work for $\mathfrak{gl}(n)$ knot homology, can't get at dimension 0, which can be minimally described as $\mathfrak{gl}(1|1)$.