Notions of positivity and the Ozsvath-Szabo concordance invariant

TL;DR

This paper explores the relationship between knot positivity types and the Ozsvath-Szabo tau invariant, revealing that for fibered knots, tau precisely characterizes strong quasipositivity, and discusses implications for knots with lens space surgeries.

Contribution

It establishes that for fibered knots, tau equals the genus if and only if the knot is strongly quasipositive, providing a new characterization within knot theory.

Findings

Tau characterizes strong quasipositivity for fibered knots.

Knots admitting lens space surgeries are strongly quasipositive.

Existence of infinite families of non-quasipositive knots.

Abstract

In this paper we examine the relationship between various types of positivity for knots and the concodance invariant tau discovered by Ozsvath and Szabo and independently by Rasmussen. The main result shows that, for fibered knots, tau characterizes strong quasipositivity. This is quantified by the statement that for K fibered, tau(K)=g(K) if and only if K is strongly quasipositive. In addition, we survey existing results regarding tau and forms of positivity and highlight several consequences concerning the types of knots which are (strongly) (quasi) positive. For instance, we show that any knot known to admit a lens space surgery is strongly quasipositive and exhibit infinite families of knots which are not quasipositive.