Secondary Upsilon invariants of knots

TL;DR

This paper introduces secondary Upsilon invariants for knots, which refine the original Upsilon invariant by providing bounds on genus and concordance genus, and can detect knots that the original invariant cannot.

Contribution

The paper defines new secondary invariants related to Upsilon that offer sharper bounds and detect more knots than the original invariant.

Findings

Secondary invariants provide bounds on knot genus and concordance genus.

Examples show secondary invariants detect knots where Upsilon vanishes.

Secondary invariants can distinguish knots not distinguished by Upsilon.

Abstract

The knot invariant Upsilon, defined by Ozsvath, Stipsicz, and Szabo, induces a homomorphism from the smooth knot concordance group to the group of piecewise linear functions on the interval [0,2]. Here we define a set of related secondary invariants, each of which assigns to a knot a piecewise linear function on [0,2]. These secondary invariants provide bounds on the genus and concordance genus of knots. Examples of knots for which Upsilon vanishes but which are detected by these secondary invariants are presented.