Topological constraints on clean Lagrangian intersections via microlocal sheaf theory

Abstract

Fix a knot $K_0$ in $\mathbb{R}^3$ and consider a Lagrangian submanifold $L$ of $T^*\mathbb{R}^3$ that is isotopic to the conormal bundle of $K_0$ by a compactly supported Hamiltonian isotopy and intersects the zero section $\mathbb{R}^3$ cleanly along a knot. In this paper, using microlocal sheaf theory and some results in $3$-manifold theory, we prove that the knot type of $K_1 := L\cap \mathbb{R}^3$ in $\mathbb{R}^3$ is strictly constrained from the knot type of $K_0$. Specifically, we deduce the existence of a surjective group homomorphism $π_1(\mathbb{R}^3\setminus K_0) \to π_1(\mathbb{R}^3\setminus K_1)$ preserving the longitude and meridian with respect to the Seifert framing. Moreover, combining with a previous work by the second author, we obtain a rigidity result which was only known for the unknot: If $K_0$ is the $(2,q)$-torus knot or the figure-eight knot, $K_1$ must have the same knot type as $K_0$.

Figures (2)

• Figure 1.1: In the left-hand side, the bullets represent the knots $K_0$ and $K_1$ in $0_{\mathbb{R}^3}=\mathbb{R}^3$. We suppose that $\varphi (T^*_{K_0}\mathbb{R}^3)$ intersects $0_{\mathbb{R}^3}$ cleanly along $K_1$ as in the right-hand side.
• Figure 1.2: We may assume that $\varphi(T^*_{K_0}\mathbb{R}^3)$ agrees with $T^*_{K_1}\mathbb{R}^3$ in a neighborhood of $0_{\mathbb{R}^3}$ (the left-hand side). The $3$-manifold $M_{\varphi}$ is diffeomorphic to the Lagrangian submanifold obtained by a Polterovich surgery on $\varphi(T^*_{K_0}\mathbb{R}^3)\cup 0_{\mathbb{R}^3}$ along $K_1$ (the right-hand side).

Theorems & Definitions (108)

• Remark 1.2
• Theorem 1.3: Theorem \ref{['thm-ml-preserve']}
• Theorem 1.4: Theorem \ref{['thm-trefoil']} and \ref{['thm-figure-8']}
• Remark 1.5
• Theorem 1.6
• Theorem 1.7: Corollary \ref{['cor-peripheral-preserve']}
• Remark 1.8
• Theorem 1.9: Theorem \ref{['thm-divide-A']}
• Theorem 1.10: Theorem \ref{['thm-KCH']}
• Remark 1.11