On the secondary Upsilon invariant

TL;DR

This paper constructs an infinite family of knots with zero Upsilon invariant but nontrivial secondary Upsilon invariants, demonstrating their independence in the smooth knot concordance group and proving a related conjecture.

Contribution

It introduces a new family of knots with vanishing Upsilon but nontrivial secondary invariants, advancing understanding of knot concordance invariants.

Findings

Knots with zero Upsilon invariant but nontrivial secondary invariants are linearly independent.

Proves a conjecture by Allen regarding secondary Upsilon invariants.

Shows the distinction between primary and secondary invariants in knot theory.

Abstract

In this paper we construct an infinite family of knots with vanishing Upsilon invariant $\Upsilon$, although their secondary Upsilon invariants $\Upsilon^2$ show that they are linearly independent in the smooth knot concordance group. We also prove a conjecture in a paper by Allen.