  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.
