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Homology of torus knots

Anton Mellit

Geometry & Topology 26 (2022) 47–70
DOI: 10.2140/gt.2022.26.47

Using the method of Elias and Hogancamp and combinatorics of toric braids from our proof of the rational shuffle conjecture, we give an explicit formula for the triply graded Khovanov–Rozansky homology (superpolynomial) of an arbitrary positive torus knot, thereby proving some of the conjectures of Aganagic and Shakirov, Cherednik, Gorsky and Negut, and Oblomkov, Rasmussen and Shende.

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torus knots, Khovanov–Rozansky homology, Dyck paths, rational shuffle conjecture
Mathematical Subject Classification 2010
Primary: 05A15, 05E05, 57M27
Received: 19 October 2018
Revised: 15 September 2020
Accepted: 5 January 2021
Published: 5 April 2022
Proposed: Ciprian Manolescu
Seconded: András I Stipsicz, Peter Ozsváth
Anton Mellit
Faculty of Mathematics
University of Vienna