Duke Mathematical Journal

Torus knots and the rational DAHA

Eugene Gorsky, Alexei Oblomkov, Jacob Rasmussen, and Vivek Shende

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Abstract

We conjecturally extract the triply graded Khovanov–Rozansky homology of the (m,n) torus knot from the unique finite-dimensional simple representation of the rational DAHA of type A, rank n1, and central character m/n. The conjectural differentials of Gukov, Dunfield, and the third author receive an explicit algebraic expression in this picture, yielding a prescription for the doubly graded Khovanov–Rozansky homologies. We match our conjecture to previous conjectures of the first author relating knot homology to q,t-Catalan numbers and to previous conjectures of the last three authors relating knot homology to Hilbert schemes on singular curves.

Article information

Source
Duke Math. J., Volume 163, Number 14 (2014), 2709-2794.

Dates
First available in Project Euclid: 31 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1414762069

Digital Object Identifier
doi:10.1215/00127094-2827126

Mathematical Reviews number (MathSciNet)
MR3273582

Zentralblatt MATH identifier
1318.57010

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 16G99: None of the above, but in this section 14H20: Singularities, local rings [See also 13Hxx, 14B05] 05A99: None of the above, but in this section

Keywords
Khovanov homology HOMFLY polynomial rational DAHA rational Cherednik algebra torus knot Hilbert scheme

Citation

Gorsky, Eugene; Oblomkov, Alexei; Rasmussen, Jacob; Shende, Vivek. Torus knots and the rational DAHA. Duke Math. J. 163 (2014), no. 14, 2709--2794. doi:10.1215/00127094-2827126. https://projecteuclid.org/euclid.dmj/1414762069


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