Topological constraints on clean Lagrangian intersections via microlocal sheaf theory
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914407021281280 |
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| author | Asano, Tomohiro Okamoto, Yukihiro |
| author_facet | Asano, Tomohiro Okamoto, Yukihiro |
| contents | 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 $K_0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_17960 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Topological constraints on clean Lagrangian intersections via microlocal sheaf theory Asano, Tomohiro Okamoto, Yukihiro Symplectic Geometry Geometric Topology 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 $K_0$. |
| title | Topological constraints on clean Lagrangian intersections via microlocal sheaf theory |
| topic | Symplectic Geometry Geometric Topology |
| url | https://arxiv.org/abs/2603.17960 |