Lagrangian fillings and complicated Legendrian unknots

Abstract

An exact Lagrangian submanifold $L$ in the symplectization of standard contact $(2n-1)$-space with Legendrian boundary $\Sigma$ can be glued to itself along $\Sigma$. This gives a Legendrian embedding $\Lambda (L,L)$ of the double of $L$ into contact $(2n+1)$-space. We show that the Legendrian isotopy class of $\Lambda (L,L)$ is determined by formal data: the manifold $L$ together with a trivialization of its complexified tangent bundle. In particular, if $L$ is a disk then $\Lambda (L,L)$ is the Legendrian unknot.

8 pages. An assumption on regularity of Lagrangian fillings was removed following a suggestion by Yang Huang. Version contains Emmy Murphy's proof of looseness of Legendrian spheres obtained by gluing two distinct disk fillings. (These spheres were originally claimed elsewhere by the second author to be non-loose.)

References in corpus (1)

• Non-loose Legendrian spheres with trivial Contact Homology DGA

Cited by in corpus (1)

• Existence of coLegendrians in contact $5$-manifolds