Characterization of positive links and the $s$-invariant for links

TL;DR

This paper characterizes positive links using strong quasipositivity, homogeneity, and the $s$-invariant, and explores the properties of almost positive links, including their $s$-invariants and potential strong quasipositivity.

Contribution

It provides a new characterization of positive links and investigates the $s$-invariants of almost positive links, proposing that all almost positive links may be strongly quasipositive.

Findings

Positive links are characterized by strong quasipositivity, homogeneity, and the $s$-invariant.

Almost positive links have specific $s$-invariants and are not homogeneous.

All almost positive links might be strongly quasipositive.

Abstract

We characterize positive links in terms of strong quasipositivity, homogeneity and the value of Rasmussen, Beliakova and Wehrli's $s$-invariant. We also study almost positive links, in particular, determine the $s$-invariants of almost positive links. This result suggests that all almost positive links might be strongly quasipositive. On the other hand, it implies that almost positive links are never homogeneous links.