Upsilon type concordance invariants

TL;DR

This paper introduces a new family of knot concordance invariants derived from convex regions in the plane, unifying and extending existing invariants like Rasmussen's and Ozsváth-Stipsicz-Szabó's, with applications to classifying knots.

Contribution

It defines a general framework for Upsilon-type invariants based on convex regions, encompassing known invariants and introducing new secondary invariants for broader knot analysis.

Findings

Computed invariants for alternating and torus knots.

Obstructed concordances to Floer thin and algebraic knots.

Unified various existing invariants within a single framework.

Abstract

To a region $C$ of the plane satisfying a suitable convexity condition we associate a knot concordance invariant $\Upsilon^C$. For appropriate choices of the domain this construction gives back some known knot Floer concordance invariants like Rasmussen's $h_i$ invariants, and the Ozsv\' ath-Stipsicz-Szab\' o upsilon invariant. Furthermore, to three such regions $C$, $C^+$ and $C^- $ we associate invariants $\Upsilon_{C^\pm, C}$ generalising Kim-Livingston secondary invariant. We show how to compute these invariants for some interesting classes of knots (including alternating and torus knots), and we use them to obstruct concordances to Floer thin knots and algebraic knots.