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arXiv · 1207.4523

Torus knots and the rational DAHA

Eugene Gorsky·Alexei Oblomkov·Jacob Rasmussen·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 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 of the last three authors relating knot homology to Hilbert schemes on singular curves.

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BibTeXRIS

Eugene Gorsky, Alexei Oblomkov, Jacob Rasmussen, Vivek Shende. 2012-07-18. Torus knots and the rational DAHA. https://doi.org/10.1215/00127094-2827126

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