Persistence of unknottedness of clean Lagrangian intersections

Let $Q_0$ and $Q_1$ be two Lagrangian spheres in a $6$-dimensional symplectic manifold. Assume that $Q_0$ and $Q_1$ intersect cleanly along a circle that is unknotted in both $Q_0$ and $Q_1$. We prove that there is no nearby Hamiltonian isotopy of $Q_0$ and $Q_1$ to a pair of Lagrangian spheres meeting cleanly along a circle that is knotted in either component. The proof is based on a classification for spherical Lagrangians in a Stein neighborhood of the union $Q_0\cup Q_1$ and the deep result that lens space rational Dehn surgeries characterizes the unknot.
Comment: 46 pages, 18 figures. Comments welcome