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Curved Rickard complexes and link homologies

  • Sabin Cautis EMAIL logo , Aaron D. Lauda und Joshua Sussan
Veröffentlicht/Copyright: 11. Februar 2020

Abstract

Rickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed [3] [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology, Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y-ification of Khovanov–Rozansky homology, preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves IV, colored links, Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper).

Award Identifier / Grant number: DMS-1255334

Award Identifier / Grant number: DMS-1664240

Award Identifier / Grant number: DMS-1807161

Funding source: Simons Foundation

Award Identifier / Grant number: 516673

Funding statement: Sabin Cautis is supported by an NSERC Discovery grant. Aaron D. Lauda is partially supported by the NSF grants DMS-1255334 and DMS-1664240. Joshua Sussan is partially supported by the NSF grant DMS-1807161, PSC-CUNY Award 61028-00 49, and Simons Foundation Collaboration Grant 516673.

Acknowledgements

The authors are grateful to Matt Hogancamp for many illuminating discussions on y-ification. Sabin Cautis thanks Eugene Gorsky for taking the time to elaborate on his recent work and Joshua Sussan would like to thank Shotaro Makisumi for explaining his work related to [17].

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Received: 2019-03-22
Revised: 2019-11-15
Published Online: 2020-02-11
Published in Print: 2020-12-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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