Slice-torus link invariants, combinatorial invariants, and positivity conditions

TL;DR

This paper establishes necessary conditions for links to be concordant to quasi-positive, positive, or braid closures, providing characterizations and criteria for positive links, and compiling a comprehensive table of small positive links.

Contribution

It introduces new combinatorial invariants and criteria for identifying positive links and their properties, including a complete classification for links with fewer than 8 crossings.

Findings

Characterization of positive links with unlinking number 1 and 2

A combinatorial criterion for positive braid closures

A complete table of positive and positive-braid prime links with fewer than 8 crossings

Abstract

We prove some necessary conditions for a link to be either concordant to a quasi-positive link, quasi-positive, positive, or the closure of a positive braid. The main applications of our results are a characterisation of positive links with unlinking number $1$ and $2$, and a combinatorial criterion to test if a positive link is the closure of a positive braid. Finally, we compile a table of all positive and positive-braid prime links with less than $8$ crossings.