Persistence of unknottedness of clean Lagrangian intersections

TL;DR

This paper proves that the unknottedness of clean Lagrangian intersections in a 6-dimensional symplectic manifold is preserved under small Hamiltonian perturbations, using classification of Lagrangian summands and Dehn surgery results.

Contribution

It establishes the persistence of unknottedness in Lagrangian intersections under Hamiltonian isotopy, answering a question posed by Smith.

Findings

Unknotted intersections cannot be perturbed into knotted ones.

Classification of Lagrangian summands is key to the proof.

Lens space Dehn surgeries characterize the unknot.

Abstract

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, answering a question of Smith. The proof is based on a classification of the spherical summands in the prime decomposition of an exact Lagrangian in the Stein neighborhood of the union $Q_0\cup Q_1$ and the deep result that lens space rational Dehn surgeries characterize the unknot.