On the Upsilon invariant of cable knots

Abstract

In this paper, we study the behavior of ${{\rm Y}}_K(t)$ under the cabling operation, where ${{\rm Y}}_K(t)$ 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 ${{\rm Y}}_K(t)$ and ${{\rm Y}}_{K_{p,q}}(t)$, which generalizes the inequalities of Hedden and Van Cott on the Ozsváth-Szabó $\tau$-invariant. As applications, we give a computation of ${{\rm Y}}_{(T_{2,-3}{)}_{2,2n+1}}(t)$ for $n\ge 8$, and we also show that the set of iterated $(p,1)$-cables of $W{h}^{+}(T_{2,3})$ for any $p\ge 2$ span an infinite-rank summand of topologically slice knots.

arXiv commentVersion 2: a new application is added

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Cited by in corpus (5)

• Concordance homomorphisms from knot Floer homology
• On the Upsilon invariant and satellite knots
• Cable knots are not thin
• On the Upsilon invariant of fibered knots and right-veering open books
• Linear independence of cables in the knot concordance group