The Superpolynomial for Knot Homologies

Abstract

We propose a framework for unifying the sl(N) Khovanov-Rozansky homology (for all N) with the knot Floer homology. We argue that this unification should be accomplished by a triply graded homology theory which categorifies the HOMFLY polynomial. Moreover, this theory should have an additional formal structure of a family of differentials. Roughly speaking, the triply graded theory by itself captures the large N behavior of the sl(N) homology, and differentials capture nonstable behavior for small N, including knot Floer homology. The differentials themselves should come from another variant of sl(N) homology, namely the deformations of it studied by Gornik, building on work of Lee.While we do not give a mathematical definition of the triply graded theory, the rich formal structure we propose is powerful enough to make many non-trivial predictions about the existing knot homologies that can then be checked directly. We include many examples where we can exhibit a likely candidate for the triply graded theory, and these demonstrate the internal consistency of our axioms. We conclude with a detailed study of torus knots, developing a picture which gives new predictions even for the original sl(2) Khovanov homology.

Link Homologies and the Refined Topological Vertex

“…This conjecture led to a number of predictions regarding the structure of sl(N ) knot homologies, in particular to the triply-graded knot homology categorifying the HOM-FLY polynomial [9,16], see also [10]. However, a direct test of this conjecture and computation of homological link invariants from string theory was difficult due to lack of techniques suitable for calculating degeneracies of BPS states in the physical setup.…”

“…which is exactly the superpolynomial of the unknot [9]. It is interesting to note that for generic representations the partition function for the unknot depends on both parameters q and t, whose interpretation we are currently investigating [31].…”

“…[17], and we use the variables (a, q, t) when we discuss link homologies, cf. [9]. The two sets of variables are related as follows:…”

“…On the other hand, a physical interpretation of link homologies was first proposed in [8] and further developed in [9,10]. The interpretation involves BPS states in the context of physical interpretation of open topological string amplitudes [11].…”

Three-dimensional extensions of the Alday-Gaiotto-Tachikawa relation

“…The last line of this table describes an extension from (quantum) groups to MacDonald characters which leads to a one-parametric deformation (t-deformation) of (13), to superpolynomials [22,23] involving the Khovanov homology [24]. Further extension from MacDonald to the Askey-Wilson-Kerov level remains untouched so far.…”