The Superpolynomial for Knot Homologies

By Nathan M. Dunfield, Sergei Gukov, Jacob Rasmussen. Experimental Mathematics, 2005

317 citations recorded by the source corpus.

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 that 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 nontrivial predictions about the existing knot homologies that can then be checked directly. We include many examples in which 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 that gives new predictions even for the original sl(2) Khovanov homology.

Subfields: Geometry and Topology, Mathematical Physics, Neurology

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Citations in the corpus

Khovanov-Rozansky Homology and Topological Strings (2004, methodology): “[39] for independent computations of the Khovanov-Rozansky homology in some of the examples considered here (see also [38]).”
Torus knots and the rational DAHA (2012, methodology): “categoriﬁcations: PK(qN,q,t) 6= PN,K(q,t). However, there is a spectral sequence HK → HN,K constructed by the third author [Ras06], which establishes part of a richer conjectural pictured advanced in [DGR06]. In all known cases, the spectral sequence converges after the ﬁrst differential. Let Kbe a ”
The universal sl3–link homology (2006, result): “[12] M. Mackaay, P. Turner, P. Vaz, A remark on Rasmussen’s invar ant of knots, preprint 2005, math.”
Holomorphic disks, link invariants and the multi-variable Alexander polynomial (2008, result): “It is interesting to compare this with the recent categorification of the HOMFLY polynomial [10], see also [8], [9], [ 17 ].”
Knot homology groups from instantons (2008, background): “The situation is reminiscent of the large-N stabilization for Khovanov-Rozansky homology that is discussed in [10, 37].”
The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link (2012, methodology): “It was conjectured in [20] that Proposition 6 from the introduction says that the analogous statement holds for P alg .”
A survey of Heegaard Floer homology (2013, background): “Note that Rasmussen introduced the s-invariant using Khovanov homology, which also gives a lower bound on the 4-ball genus, as |spKq| ď 2g˚pKq. Rasmussen conjectured that spKq “ 2τpKq, which was then disproved by Hedden and Ording [29].”
Jones polynomials of torus knots via DAHA (2011, methodology): “The focus is on the 3-parametric superpolynomials [1, 11, 17, 18, 24, 31].”
The Ozsváth-Szabó and Rasmussen Concordance Invariants are not Equal (2005, background): “In fact, the above knots provide further counterexamples, as was shared with us by Jake Rasmussen, who used Bar-Natan’s program [1] for computing Khovanov homology to calculate s for the knots in the above family which are not covered by Livingston and Naik’s result.”
Deformations of colored sl(N) link homologies via foams (2015, background): “and physical reasons suggest that the type A link homology package carries a very rigid structure, which is only partially visible on the decategoriﬁed level of Reshetikhin-Turaev slN invariants, see [DGR], [GW], [GS], [GGS] and [GNSS]. One feature of this structure is the stabilization of slN link ”
Fivebranes and 3-manifold homology (2016, methodology): “t space, which is the Hilbert space of a closed string dual and which reduces for each integer Nto the 3-manifold homology (1.1). This structural property parallels what was found for knots and links [7,10,29]. Appendices contain useful supplementary material. { 7 { 2. Categorication of a 2d A-model”
Homological algebra of knots and BPS states (2011, methodology): “ry whose Poincar´e polynomial is equal to PSr(K)(a,q,t), and whose Euler characteristic is equal to the normalized Srcolored two-variable HOMFLY polynomial. As in the case of ordinary HOMFLY homology [3] (that, in fact, corresponds to r = 1) and in the case of Kauﬀman homology [6], this triply-grade”
DAHA and iterated torus knots (2014, methodology): “A similar connection is also expected with the physics superpolynomials associated with the BPS states (see eg [15; 1; 16]) and those related to the Hilbert schemes of plane curve singularities and rational DAHA; see [33; 23; 22] and references therein.”
Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology (2016, background): “In the spirit of the above conjecture, contraction with s1,1 may give rise to a differential on HHH whose homology is knot Floer homology, as conjectured in [21].”
AN INTRODUCTION TO KNOT FLOER HOMOLOGY (2014, unknown): “As mentioned in the introduction, Dunﬁeld, Gukov and Rasmussen [DGR06] conjectured the existence of a spectral sequence with E 2 page the Khovanov-Rozansky homology H KR ( K ), and converging to knot Floer homology.”
Khovanov-Rozansky homology of two-bridge knots and links (2005, background): “For N > 4, we find (86) P(HN (41)) = [N ] + [N − 1](q t + qt + qt + qt(2)) thus validating the prediction made in equation (61) of [5].”