On knot types of clean Lagrangian intersections in $T^*\mathbb{R}^3$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916708025892864 |
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| author | Okamoto, Yukihiro |
| author_facet | Okamoto, Yukihiro |
| contents | Let $K_0$ and $K$ be knots in $\mathbb{R}^3$. Suppose that by a compactly supported Hamiltonian isotopy on $T^*\mathbb{R}^3$, the conormal bundle of $K_0$ is isotopic to a Lagrangian submanifold which intersects the zero section cleanly along $K$. In this paper, we prove some constraints on the pair of knot types of $K_0$ and $K$. One example is that if $K_0$ is the unknot, then $K$ is also the unknot. We also consider some cases where $K_0$ and $K$ have specific knot types, such as torus knots and connected sums of trefoil knots. The key step is finding a DGA map between the Chekanov-Eliashberg DGAs of the unit conormal bundles of knots. The main results are deduced from a relation between the augmentation varieties of $K_0$ and $K$ determined by these DGAs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_09912 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On knot types of clean Lagrangian intersections in $T^*\mathbb{R}^3$ Okamoto, Yukihiro Symplectic Geometry Geometric Topology Primary 53D42, Secondary 53D40, 57K10 Let $K_0$ and $K$ be knots in $\mathbb{R}^3$. Suppose that by a compactly supported Hamiltonian isotopy on $T^*\mathbb{R}^3$, the conormal bundle of $K_0$ is isotopic to a Lagrangian submanifold which intersects the zero section cleanly along $K$. In this paper, we prove some constraints on the pair of knot types of $K_0$ and $K$. One example is that if $K_0$ is the unknot, then $K$ is also the unknot. We also consider some cases where $K_0$ and $K$ have specific knot types, such as torus knots and connected sums of trefoil knots. The key step is finding a DGA map between the Chekanov-Eliashberg DGAs of the unit conormal bundles of knots. The main results are deduced from a relation between the augmentation varieties of $K_0$ and $K$ determined by these DGAs. |
| title | On knot types of clean Lagrangian intersections in $T^*\mathbb{R}^3$ |
| topic | Symplectic Geometry Geometric Topology Primary 53D42, Secondary 53D40, 57K10 |
| url | https://arxiv.org/abs/2305.09912 |