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

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.