Duke Mathematical Journal

Torus knots and the rational DAHA

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

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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.

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

First available in Project Euclid: 31 October 2014

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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

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


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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