Topological constraints on clean Lagrangian intersections via microlocal sheaf theory

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 $K_0$.