Positive Knots and Lagrangian Fillability

TL;DR

This paper investigates the connection between positive knots and Lagrangian fillability, demonstrating that positive knots can have Legendrian representatives with exact Lagrangian fillings and exploring the independence of fillability and strong quasi-positivity.

Contribution

It establishes that positive knots admit Legendrian representatives with exact Lagrangian fillings and shows the independence of fillability and strong quasi-positivity.

Findings

Positive knots have Legendrian representatives with exact Lagrangian fillings.

Every Legendrian knot with an exact Lagrangian filling is quasi-positive.

Strong quasi-positivity and fillability are independent conditions.

Abstract

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.