Mathematical Research Letters

Volume 29 (2022)

Number 2

On the Upsilon invariant of fibered knots and right-veering open books

Pages: 421 – 444



Dongtai He (Shanghai Center for Mathematical Science, Fudan University, Shanghai, China)

Diana Hubbard (Department of Mathematics, Brooklyn College, Brooklyn, New York, U.S.A.)

Linh Truong (Department of Mathematics, University of Michigan, Ann Arbor, Mich., U.S.A.)


We give a sufficient condition using the Ozsváth–Stipsicz–Szabó concordance invariant Upsilon for the monodromy of the open book decomposition of a fibered knot to be right-veering. As an application, we generalize a result of Baker on ribbon concordances between fibered knots. Following Baker, we conclude that either fibered knots $K$ in $S^3$ satisfying that $\mathcal{Υ}^\prime (t) = -g(K)$ for some $t \in [0, 1)$ are unique in their smooth concordance classes or there exists a counterexample to the Slice-Ribbon Conjecture.

The main theorem of this paper grew out of ideas from the first author’s PhD thesis [10]. We thank Eli Grigsby for suggesting and encouraging this collaboration and her interest in this project. We also thank Jen Hom and Olga Plamenevskaya for thoughtful comments. The second and third authors would like to thank the organizers of the 2019 Women in Symplectic and Contact Geometry and Topology (WiSCon) workshop at ICERM, where some of this work took place, and are grateful to ICERM and to the AWM for supporting WiSCon via the AWM ADVANCE grant NSF-HRD 1500481. The second author was supported in part by an AMS-Simons travel grant. The third author was partially supported by NSF grant DMS-1606451 and the Institute for Advanced Study.

Received 11 February 2020

Accepted 1 December 2020

Published 29 September 2022