On the Upsilon invariant of cable knots

TL;DR

This paper investigates how the Upsilon invariant behaves under cabling operations on knots, providing inequalities, explicit computations, and implications for the structure of topologically slice knots.

Contribution

It establishes a general inequality relating Upsilon invariants of a knot and its cables, extending previous results for the tau-invariant, and explores applications to knot concordance.

Findings

Derived a new inequality for Upsilon invariants under cabling

Computed Upsilon invariants for specific cable knots involving torus knots

Showed certain cable knots form an infinite-rank summand of topologically slice knots

Abstract

In this paper, we study the behavior of $\Upsilon_K(t)$ under the cabling operation, where $\Upsilon_K(t)$ is the knot concordance invariant defined by Ozsv\'ath, Stipsicz, and Szab\'o, associated to a knot $K\subset S^3$. The main result is an inequality relating $\Upsilon_K(t)$ and $\Upsilon_{K_{p,q}}(t)$, which generalizes the inequalities of Hedden and Van Cott on the Ozsv\'ath-Szab\'o $\tau$-invariant. As applications, we give a computation of $\Upsilon_{(T_{2,-3})_{2,2n+1}}(t)$ for $n\geq 8$, and we also 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.