Pure and Applied Mathematics Quarterly

Volume 13 (2017)

Number 3

Special Issue in Honor of Simon Donaldson

Guest Editors: Kefeng Liu, Richard Thomas, and Shing-Tung Yau

Annular Khovanov–Lee homology, braids, and cobordisms

Pages: 389 – 436

DOI: https://dx.doi.org/10.4310/PAMQ.2017.v13.n3.a2

Authors

J. Elisenda Grigsby (Department of Mathematics, Boston College, Chestnut Hill, Massachusetts, U.S.A.)

Anthony M. Licata (Mathematical Sciences Institute, Australian National University, Canberra, New South Wales, Australia)

Stephan M. Wehrli (Department of Mathematics, Syracuse University, Syracuse, New York, U.S.A.)

Abstract

We prove that the Khovanov–Lee complex of an oriented link, $L$, in a thickened annulus, $A \times I$, has the structure of a $(\mathbb{Z} \oplus \mathbb{Z})$-filtered complex whose filtered chain homotopy type is an invariant of the isotopy class of $L \subset (A \times I)$. Using ideas of Ozsváth–Stipsicz–Szabó as reinterpreted by Livingston, we use this structure to define a family of annular Rasmussen invariants that yield information about annular and non-annular cobordisms. Focusing on the special case of annular links obtained as braid closures, we use the behavior of the annular Rasmussen invariants to obtain a necessary condition for braid quasipositivity and a sufficient condition for right-veeringness.

Full Text (PDF format)

J.E.G. was partially supported by NSF CAREER award DMS-1151671.

S.M.W. was partially supported by NSF grant DMS-1111680.

Received 29 June 2017

Published 12 November 2018