Categorified Young symmetrizers and stable homology of torus links

Hogancamp, Matthew

doi:10.2140/gt.2018.22.2943

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Categorified Young symmetrizers and stable homology of torus links

Matthew Hogancamp

Geometry & Topology 22 (2018) 2943–3002

DOI: 10.2140/gt.2018.22.2943

Abstract

We show that the triply graded Khovanov–Rozansky homology of the torus link $T_{n, k}$ stabilizes as $k \to \infty$ . We explicitly compute the stable homology, as a ring, which proves a conjecture of Gorsky, Oblomkov, Rasmussen and Shende. To accomplish this, we construct complexes $P_{n}$ of Soergel bimodules which categorify the Young symmetrizers corresponding to one-row partitions and show that $P_{n}$ is a stable limit of Rouquier complexes. A certain derived endomorphism ring of $P_{n}$ computes the aforementioned stable homology of torus links.

Keywords

link homology, categorification

Mathematical Subject Classification 2010

Primary: 18G60, 57M27

References

Publication

Received: 16 March 2017

Revised: 11 January 2018

Accepted: 13 February 2018

Published: 1 June 2018

Proposed: Ciprian Manolescu

Seconded: Jim Bryan, Haynes R Miller

Authors

Matthew Hogancamp
Department of Mathematics University of Southern California Los Angeles, CA United States

Volume 22, issue 5 (2018)