Topological constraints on clean Lagrangian intersections via microlocal sheaf theory

2026/03/18 by Tomohiro Asano, Yukihiro Okamoto · 1 voice
Mathematics · #math.SG #math.GT

paper · pdf

Abstract

Fix a knot K₀ in ℝ³ and consider a Lagrangian submanifold L of T^*ℝ³ that is isotopic to the conormal bundle of K₀ by a compactly supported Hamiltonian isotopy and intersects the zero section ℝ³ 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 K₁ := L∩ ℝ³ in ℝ³ is strictly constrained from the knot type of K₀. Specifically, we deduce the existence of a surjective group homomorphism π₁(ℝ³∖ K₀) → π₁(ℝ³∖ K₁) 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 K₀ is the (2,q)-torus knot or the figure-eight knot, K₁ must have the same knot type as K₀.

Citations

• Persistence of unknottedness of clean Lagrangian intersections
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• Lagrangian cobordism functor in microlocal sheaf theory I
• Sheaf quantization in Weinstein symplectic manifolds
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• Centralizers in 3-manifold groups
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• Sheaf quantization of Hamiltonian isotopies and applications to non displaceability problems
• Effect of Legendrian Surgery
• Epimorphisms between 2–bridge link groups
• Rational Symplectic Field Theory over Z₂ for exact Lagrangian cobordisms
• Twisted Alexander polynomials and surjectivity of a group homomorphism
• Framed knot contact homology
• Non-triviality of the A-polynomial for knots in S³
• Witten's conjecture and Property P
• Orderable 3-manifold groups
• Orderable 3-manifold groups

Discussions

• Tomohiro Asano, Yukihiro Okamoto: Topological constraints on clean Lagrangian intersections via microlocal sheaf theory https://arxiv.org/abs/2603.17960 https://arxiv.org/pdf/2603.17960 https://arxiv. [bsky, 0 points, 0 comments]

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