A note on the concordance invariant epsilon

TL;DR

This paper investigates the relationships among three smooth concordance invariants—Upsilon, phi, and epsilon—by constructing an infinite family of knots with specific invariant properties, highlighting differences in their detection capabilities.

Contribution

It introduces an infinite family of linearly independent knots where Upsilon and phi are zero but epsilon is nonzero, demonstrating the distinctness of these invariants.

Findings

Constructed an infinite family of knots with specific invariant properties

Showed that epsilon can be nonzero even when Upsilon and phi are zero

Highlighted the independence of the epsilon invariant from Upsilon and phi

Abstract

We compare the smooth concordance invariants Upsilon, phi and epsilon. Previous work gave examples of knots with one of the Upsilon and phi invariants zero but the epsilon invariant nonzero. We build an infinite family of linearly independent knots with both the Upsilon and phi invariants zero but the epsilon invariant nonzero.