#### Volume 21, issue 3 (2021)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
On the Upsilon invariant of cable knots

### Wenzhao Chen

Algebraic & Geometric Topology 21 (2021) 1075–1092
##### Abstract

We study the behavior of ${\Upsilon }_{K}\left(t\right)$ under the cabling operation, where ${\Upsilon }_{K}\left(t\right)$ 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 ${\Upsilon }_{K}\left(t\right)$ and ${\Upsilon }_{{K}_{p,q}}\left(t\right)$, where ${K}_{p,q}$ denotes the $\left(p,q\right)$–cable of $K$. This result generalizes the inequalities of Hedden and Van Cott on the Ozsváth–Szabó $\tau$–invariant. As applications, we give a computation of ${\Upsilon }_{{\left({T}_{2,-3}\right)}_{2,2n+1}}\left(t\right)$ for $n\ge 8$, and we show that the set of iterated $\left(p,1\right)$–cables of ${Wh}^{+}\left({T}_{2,3}\right)$ for any $p\ge 2$ 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