A note on the concordance invariants epsilon and upsilon

TL;DR

This paper compares two knot concordance invariants, upsilon and epsilon, highlighting their differences through an example where upsilon is zero but epsilon is non-zero, revealing nuances in their behavior.

Contribution

It provides a comparison between the invariants upsilon and epsilon, including an explicit example illustrating their distinct values.

Findings

Upsilon can be zero while epsilon is non-zero for the same knot.

The invariants capture different aspects of knot concordance.

The paper clarifies the relationship between two important knot invariants.

Abstract

Ozsvath-Stipsicz-Szabo recently defined a one-parameter family, upsilon of K at t, of concordance invariants associated to the knot Floer complex. We compare their invariant to the {-1, 0, 1}-valued concordance invariant epsilon, which is also associated to the knot Floer complex. In particular, we give an example of a knot K with upsilon uniformly equal to zero but epsilon non-zero.