A note on the concordance invariants Upsilon and phi

TL;DR

This paper explores the relationship between the Upsilon and phi concordance invariants, providing examples of knots with specific invariant properties and establishing their linear independence in the concordance group.

Contribution

It demonstrates the existence of infinitely many knots with particular combinations of Upsilon and phi invariants and offers a recursive formula for phi of torus knots.

Findings

Existence of infinitely many knots with zero Upsilon but nonzero phi invariants.

Existence of infinitely many knots with zero phi but nonzero Upsilon invariants.

A recursive formula for calculating the phi invariant of torus knots.

Abstract

Dai, Hom, Stoffregen and Truong defined a family of concordance invariants $\varphi_j$. The example of a knot with zero Upsilon invariant but nonzero epsilon invariant previously given by Hom also has nonzero phi invariant. We show there are infinitely many such knots that are linearly independent in the smooth concordance group. In the opposite direction, we build infinite families of linearly independent knots with zero phi invariant but nonzero Upsilon invariant. We also give a recursive formula for the phi invariant of torus knots.