Notes on the knot concordance invariant Upsilon

TL;DR

This paper discusses the Upsilon invariant in knot theory, providing a new approach to defining it and exploring its fundamental properties related to knot genus and concordance.

Contribution

It introduces a novel method for defining the Upsilon invariant and establishes its basic properties concerning knot genus and concordance.

Findings

Upsilon is related to knot 3-genus, 4-genus, and concordance genus

Provides a new approach to defining Upsilon

Establishes fundamental properties of Upsilon

Abstract

The knot concordance invariant Upsilon, recently defined by Ozsvath, Stipsicz, and Szabo, takes values in the group of piecewise linear functions on the closed interval [0,2]. This paper presents a description of one approach to defining Upsilon and of proving its basic properties related to the knot 3-genus, 4-genus, and concordance genus.