Persistence of unknottedness of clean Lagrangian intersections

TL;DR

This work studies rigidity of clean Lagrangian intersections in 6-dimensional symplectic manifolds by modeling a local situation with double bubble plumbings $W_k$. It computes the fiberwise wrapped Fukaya category of $W_k$, identifies it with a Ginzburg dg algebra $ ext{G}_k$, and establishes a Calabi–Yau framework that admits cyclic quasi-dilations, enabling exact-Lagrangian classification. A key outcome is a derived equivalence between the wrapped Fukaya category of $W_k$ and the relative coherent category of its mirror $Y_k$, linking symplectic topology to singularity theory and mirror symmetry. Using this setup, the authors rule out the existence of knotted clean intersections under nearby Hamiltonian isotopies of the core spheres, by combining a detailed prime-summand analysis with dilation-based obstructions. Consequently, for two Lagrangian spheres intersecting along an unknot in a 6-manifold, the knot type of the intersection is preserved under nearby Hamiltonian isotopies, providing a negative answer to Smith’s question in this setting.

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.

Figures (13)

• Figure 1: The projections of $Q_0$, $Q_1$, and the arcs $\gamma_0$, $\gamma_1$.
• Figure 2: The arcs $\gamma_{1,t}$ and $\gamma_{1,t'}$ with a shaded bigon contributing to the Floer differential.
• Figure 3: The arcs $\gamma_{0,t}$ and $\gamma_{0,t'}$ with a shaded bigon contributing to the Floer differential.
• Figure 4: The arcs $\gamma_{0,t}$, $\gamma_{1,t'}$ and $\gamma_{0,t"}$, and shaded triangles contributing to the Floer product \ref{['eq:tp1']}.
• Figure 5: The arcs $\gamma_{0,t}$, $\gamma_{0,t'}$ and $\gamma_{1,t"}$, and a shaded triangle contributing to the Floer product \ref{['eq:LLT']}.

Theorems & Definitions (80)

• Theorem 1.1
• Theorem 1.3: \ref{['theorem:main']}
• Remark 1.4
• Theorem 1.5: \ref{['theorem:MS']}
• Remark 1.6
• Corollary 1.7
• proof
• Lemma 1.8
• Remark 1.9
• Theorem 1.10: \ref{['theorem:ginzburg']}