Lagrangian fillings and complicated Legendrian unknots

An exact Lagrangian submanifold $L$ in the symplectization of standard contact $(2n-1)$-space with Legendrian boundary $\mathrm{\Sigma }$ can be glued to itself along $\mathrm{\Sigma }$. This gives a Legendrian embedding $\mathrm{\Lambda }(L,L)$ of the double of $L$ into contact $(2n+1)$-space. We show that the Legendrian isotopy class of $\mathrm{\Lambda }(L,L)$ 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 $\mathrm{\Lambda }(L,L)$ is the Legendrian unknot.