Topological constraints on clean Lagrangian intersections from $\mathbb{Q}$-valued augmentations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908874618961920 |
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| author | Okamoto, Yukihiro |
| author_facet | Okamoto, Yukihiro |
| contents | Let $K$ be a knot in $\mathbb{R}^3$ which has the $(2,q)$-torus knot for $q\neq \pm 1$ or the figure-eight knot as a component of connected sum. For its conormal bundle $L_K$ in $T^*\mathbb{R}^3$, we show that there is no compactly supported Hamiltonian diffeomorphism $φ$ on $T^*\mathbb{R}^3$ such that $φ(L_K)$ intersects the zero section $\mathbb{R}^3$ cleanly along the unknot in $\mathbb{R}^3$. Using symplectic field theory, the proof is reduced to studying the augmentation variety $V_{\mathbf{k}}(K)$ of $K$ over a filed $\mathbf{k}$. The key point of this paper is finding an algebraic constraint on $V_{\mathbf{k}}(K)$ which is valid only when $\mathbf{k}$ is not algebraically closed, and the proof is completed by some arithmetic argument with $\mathbf{k}=\mathbb{Q}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_00330 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Topological constraints on clean Lagrangian intersections from $\mathbb{Q}$-valued augmentations Okamoto, Yukihiro Symplectic Geometry Geometric Topology 53D42, 53D40, 57K10 Let $K$ be a knot in $\mathbb{R}^3$ which has the $(2,q)$-torus knot for $q\neq \pm 1$ or the figure-eight knot as a component of connected sum. For its conormal bundle $L_K$ in $T^*\mathbb{R}^3$, we show that there is no compactly supported Hamiltonian diffeomorphism $φ$ on $T^*\mathbb{R}^3$ such that $φ(L_K)$ intersects the zero section $\mathbb{R}^3$ cleanly along the unknot in $\mathbb{R}^3$. Using symplectic field theory, the proof is reduced to studying the augmentation variety $V_{\mathbf{k}}(K)$ of $K$ over a filed $\mathbf{k}$. The key point of this paper is finding an algebraic constraint on $V_{\mathbf{k}}(K)$ which is valid only when $\mathbf{k}$ is not algebraically closed, and the proof is completed by some arithmetic argument with $\mathbf{k}=\mathbb{Q}$. |
| title | Topological constraints on clean Lagrangian intersections from $\mathbb{Q}$-valued augmentations |
| topic | Symplectic Geometry Geometric Topology 53D42, 53D40, 57K10 |
| url | https://arxiv.org/abs/2505.00330 |