Persistence of Unknottedness of Lagrangian Intersections

The double bubble plumbing, first studied by Smith and Wemyss, is a Stein neighborhood of two Lagrangian 3-spheres intersecting cleanly along an unknotted circle in some 6-dimensional symplectic manifold. Depending on the identification of the normal bundle of the unknot, there are infinitely many such Stein neighborhoods. We prove that there is no Hamiltonian isotopy of the Lagrangian spheres in any of these Stein neighborhoods so that they become two spheres intersecting along a circle which is knotted in either component, contradicting what happens under smooth isotopies. The proof uses the exact Calabi-Yau structures on the wrapped Fukaya categories to classify spherical Lagrangians in double bubble plumbings. This is joint work with Johan Asplund.