Positive knots and Lagrangian fillability

Hayden, Kyle; Sabloff, Joshua

doi:10.1090/S0002-9939-2014-12365-3

Positive knots and Lagrangian fillability
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by Kyle Hayden and Joshua M. Sabloff PDF

Proc. Amer. Math. Soc. 143 (2015), 1813-1821 Request permission

Abstract:

This paper explores the relationship between the existence of an exact embedded Lagrangian filling for a Legendrian knot in the standard contact $\mathbb {R}^3$ 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.

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