This paper explores the relationship between the existence of an exact
embedded Lagrangian filling for a Legendrian knot in the standard contact
$\rr^3$ and the hierarchy of positive, strongly quasi-positive, and
quasi-positive knots. On one hand, results of Eliashberg and especially Boileau
and Orevkov show that every Legendrian knot with an exact, embedded Lagrangian
filling is quasi-positive. On the other hand, we show that if a knot type is
positive, then it has a Legendrian representative with an exact embedded
Lagrangian filling. Further, we produce examples that show that strong
quasi-positivity and fillability are independent conditions.
Arxiv
