Almost positive links are strongly quasipositive

TL;DR

This paper proves that links with diagrams having a single negative crossing are strongly quasipositive, characterizes diagrams with quasipositive canonical surfaces, and confirms a conjecture relating strong quasipositivity to the Bennequin inequality for certain knots.

Contribution

It establishes that links with one negative crossing are strongly quasipositive and provides complete characterizations of diagrams with quasipositive canonical surfaces.

Findings

Prime knots up to 13 crossings are strongly quasipositive.

Links with a single negative crossing are strongly quasipositive.

Confirmed the conjecture relating strong quasipositivity and the Bennequin inequality for certain knots.

Abstract

We prove that any link admitting a diagram with a single negative crossing is strongly quasipositive. This answers a question of Stoimenow's in the (strong) positive. As a second main result, we give simple and complete characterizations of link diagrams with quasipositive canonical surface (the surface produced by Seifert's algorithm). As applications, we determine which prime knots up to 13 crossings are strongly quasipositive, and we confirm the following conjecture for knots that have a canonical surface realizing their genus: a knot is strongly quasipositive if and only if the Bennequin inequality is an equality.