On the Upsilon invariant of cable knots

Chen, Wenzhao

doi:10.2140/agt.2021.21.1075

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On the Upsilon invariant of cable knots

Wenzhao Chen

Algebraic & Geometric Topology 21 (2021) 1075–1092

DOI: 10.2140/agt.2021.21.1075

Abstract

We study the behavior of $ϒ_{K} (t)$ under the cabling operation, where $ϒ_{K} (t)$ is the knot concordance invariant defined by Ozsváth, Stipsicz, and Szabó, associated to a knot $K \subset S^{3}$ . The main result is an inequality relating $ϒ_{K} (t)$ and $ϒ_{K_{p, q}} (t)$ , where $K_{p, q}$ denotes the $(p, q)$ –cable of $K$ . This result generalizes the inequalities of Hedden and Van Cott on the Ozsváth–Szabó $τ$ –invariant. As applications, we give a computation of $ϒ_{{(T_{2, - 3})}_{2, 2 n + 1}} (t)$ for $n \geq 8$ , and we show that the set of iterated $(p, 1)$ –cables of ${Wh}^{+} (T_{2, 3})$ for any $p \geq 2$ span an infinite-rank summand of topologically slice knots.

Keywords

Upsilon invariant, knot concordance, cable, knot Floer homology

Mathematical Subject Classification 2010

Primary: 57M25, 57R58

References

Publication

Received: 9 August 2017

Accepted: 19 July 2020

Published: 11 August 2021

Authors

Wenzhao Chen
Max Planck Institute for Mathematics Bonn Germany

Volume 21, issue 3 (2021)