On the Upsilon invariant of cable knots

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.