Positive knots and Lagrangian fillability

Abstract

This paper explores the relationship between the existence of an exact embedded Lagrangian filling for a Legendrian knot in the standard contact R 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:1307.7683v1 [math.SG]

Obstructions to Lagrangian concordance

“…Proof. The "if" direction is due to Hayden and Sabloff [26], who showed that all positive links bound exact Lagrangian surfaces. Conversely, suppose that K bounds a Lagrangian surface Σ.…”

“…In symplectic and contact topology, there has been a great deal of recent interest in the subject of Lagrangian cobordisms between Legendrian submanifolds; see for example [1,3,5,6,8,9,14,13,25,26,42]. A key motivation is that one can construct a category whose objects are Legendrian submanifolds and whose morphisms are exact Lagrangian cobordisms, and this category fits nicely into Symplectic Field Theory [15].…”

Nonorientable Lagrangian cobordisms between Legendrian knots

“…[29],[31]). Each positive knot, 2-bridge knot, and +-adequate knot has a Legendrian representative that does not have an exact, nonorientable Langrangian endocobordism.…”

Lagrangian cobordisms via generating families: Construction and geography

“…In this section, we illustrate the possibilities of the construction in two families of examples. As noted in the introduction, deeper applications of these constructions appear in [6,8,28,38].…”