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The superpolynomial for knot homologies. (English) Zbl 1118.57012
Several link invariants which are describable in terms of skein relations have recently been categorified, i.e., expressed as the graded Poincaré polynomial of a homology theory. Examples include Khovanov homology which categorifies the Jones polynomial, Khovanov-Rozansky homology which categorifies the sl\((N)\) quantum invariant (a specialization of the HOMFLY polynomial), and knot Floer homology which categorifies the Alexander polynomial.
In this paper, the authors conjecture the existence of a single homology theory which unifies these and other categorified knot invariants, in a way reminiscent of how the HOMFLY polynomial unifies the Jones and Alexander-Conway polynomials. Though unable to define precisely this new homology theory, the authors are able to make some fairly specific predictions about its form – they predict that the theory should be triply-graded, they present a list of axioms that the differentials should satisfy, etc. Moreover, the authors are able to compute the resulting “superpolynomial” (the graded Poincaré characteristic of the conjectured homology theory) for various types of torus knots.

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57R56 Topological quantum field theories (aspects of differential topology)
57R58 Floer homology
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