On the Upsilon invariant and satellite knots

TL;DR

This paper investigates how satellite operations influence the Upsilon invariant, revealing new structures in the smooth concordance group of knots, especially regarding topologically slice knots and the Mazur satellite operator.

Contribution

It provides new results on the effect of satellite operations on the Upsilon invariant and identifies infinite rank subgroups within the concordance group.

Findings

The set _{2^i,1} is a basis for an infinite rank summand of the concordance group.

The image of the Mazur satellite operator contains an infinite rank subgroup of topologically slice knots.

Satellite operations can produce independent knots with respect to the Upsilon invariant.

Abstract

We study the effect of satellite operations on the Upsilon invariant of Ozsvath-Stipsicz-Szabo. We obtain results concerning when a knot and its satellites are independent; for example, we show that the set $\{D_{2^i,1}\}_{i=1}^\infty$ is a basis for an infinite rank summand of the group of smooth concordance classes of topologically slice knots, for D the positive clasped untwisted Whitehead double of any knot with positive tau-invariant, e.g. the right-handed trefoil. We also prove that the image of the Mazur satellite operator on the smooth knot concordance group contains an infinite rank subgroup of topologically slice knots.