Experimental Mathematics

The Superpolynomial for Knot Homologies

Nathan M. Dunfield, Sergei Gukov, and Jacob Rasmussen

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Abstract

We propose a framework for unifying the $\ssl(N)$ 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 $\ssl(N)$ homology, and differentials capture nonstable behavior for small $N$, including knot Floer homology. The differentials themselves should come from another variant of $\ssl(N)$ 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 $\ssl(2)$ Khovanov homology.

Article information

Source
Experiment. Math. Volume 15, Issue 2 (2006), 129-160.

Dates
First available in Project Euclid: 5 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.em/1175789736

Mathematical Reviews number (MathSciNet)
MR2253002

Zentralblatt MATH identifier
1118.57012

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57R56: Topological quantum field theories 57R58: Floer homology 81T30: String and superstring theories; other extended objects (e.g., branes) [See also 83E30] 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]

Keywords
HOMFLY polynomial Khovanov-Rozansky homology knot Floer homology

Citation

Dunfield, Nathan M.; Gukov, Sergei; Rasmussen, Jacob. The Superpolynomial for Knot Homologies. Experiment. Math. 15 (2006), no. 2, 129--160. https://projecteuclid.org/euclid.em/1175789736


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