## Duke Mathematical Journal

### Torus knots and the rational DAHA

#### 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 $n-1$, 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

https://projecteuclid.org/euclid.dmj/1414762069

Digital Object Identifier
doi:10.1215/00127094-2827126

Mathematical Reviews number (MathSciNet)
MR3273582

Zentralblatt MATH identifier
1318.57010

#### 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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