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 numbers 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.