Lagrangian fillings and complicated Legendrian unknots

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