Topological constraints on clean Lagrangian intersections via microlocal sheaf theory

TL;DR

This paper uses microlocal sheaf theory to establish topological constraints on Lagrangian intersections with the zero section in cotangent bundles, linking knot types via fundamental group homomorphisms and rigidity results.

Contribution

It introduces new topological restrictions on knot types of Lagrangian intersections, extending known rigidity results beyond the unknot to specific nontrivial knots.

Findings

Existence of a surjective group homomorphism between knot complements' fundamental groups.

Knot type of intersection is constrained by the original knot type.

Rigidity results for certain knots like (2,q)-torus and figure-eight knots.

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 $\pi_1(\mathbb{R}^3\setminus K_0) \to \pi_1(\mathbb{R}^3\setminus K_1)$ 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 $K_0$.