Topological constraints on clean Lagrangian intersections from ℚ-valued augmentations

2025/05/01 by Yukihiro Okamoto, Okamoto, Yukihiro · 1 voice
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Advanced Topology and Set Theory #Knot (papermaking) #Knot invariant #Knot theory #Mathematical Dynamics and Fractals #Seifert surface #Trefoil knot #Tricolorability #Unknot #math.GT #math.SG

paper · pdf · doi:10.48550/arxiv.2505.00330

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2025/05/01 · openalex created_date 2025/10/16 · openalex updated_date 2026/08/05

Abstract

Let K be a knot in ℝ³ which has the (2,q)-torus knot for q≠ ± 1 or the figure-eight knot as a component of connected sum. For its conormal bundle L_K in T^*ℝ³, we show that there is no compactly supported Hamiltonian diffeomorphism φ on T^*ℝ³ such that φ(L_K) intersects the zero section ℝ³ cleanly along the unknot in ℝ³. Using symplectic field theory, the proof is reduced to studying the augmentation variety V_k(K) of K over a filed k. The key point of this paper is finding an algebraic constraint on V_k(K) which is valid only when k is not algebraically closed, and the proof is completed by some arithmetic argument with k=ℚ.

Discussions

• Yukihiro Okamoto: A topological constraint on clean Lagrangian intersections from $\mathbb{Q}$-valued augmentations https://arxiv.org/abs/2505.00330 https://arxiv.org/pdf/2505.00330 https://arxiv.org/ [bsky, 0 points, 1 comments]

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