Lagrangian fillings and complicated Legendrian unknots

TL;DR

This paper demonstrates that certain Legendrian embeddings derived from Lagrangian fillings are classified by formal data, revealing that even complex constructions like doubles of Lagrangians can be understood through topological trivializations.

Contribution

It establishes that the Legendrian isotopy class of embeddings from doubled Lagrangians is determined solely by formal topological data, simplifying their classification.

Findings

Legendrian embeddings from doubled Lagrangians are classified by formal data.

If the Lagrangian is a disk, the resulting Legendrian is the unknot.

The classification depends on trivializations of the complexified tangent bundle.

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.