An algorithm to Legendrian realize a curve on a ribbon surface
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917395132579840 |
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| author | Stenhede, Eric |
| author_facet | Stenhede, Eric |
| contents | We give an explicit algorithm to Legendrian realize a homologically nontrivial simple closed curve on a ribbon surface of a Legendrian graph in the standard contact structure $(\mathbb{R}^3,ξ_{\rm st})$. As an application, we obtain an algorithm that converts an abstract open book whose monodromy is written as a product of Dehn twists along homologically nontrivial curves into a contact surgery diagram for the supported contact manifold. Along the way, we also record a uniqueness statement which is implicit in earlier work but, to our knowledge, was never written in the form needed here: any two Legendrian realizations of the same curve on a ribbon surface are Legendrian isotopic, and likewise for Legendrian knots lying on pages of open books and representing the same isotopy class on the page. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_08010 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An algorithm to Legendrian realize a curve on a ribbon surface Stenhede, Eric Geometric Topology Symplectic Geometry 57K33 (Primary) 57K10, 53D10 (Secondary) We give an explicit algorithm to Legendrian realize a homologically nontrivial simple closed curve on a ribbon surface of a Legendrian graph in the standard contact structure $(\mathbb{R}^3,ξ_{\rm st})$. As an application, we obtain an algorithm that converts an abstract open book whose monodromy is written as a product of Dehn twists along homologically nontrivial curves into a contact surgery diagram for the supported contact manifold. Along the way, we also record a uniqueness statement which is implicit in earlier work but, to our knowledge, was never written in the form needed here: any two Legendrian realizations of the same curve on a ribbon surface are Legendrian isotopic, and likewise for Legendrian knots lying on pages of open books and representing the same isotopy class on the page. |
| title | An algorithm to Legendrian realize a curve on a ribbon surface |
| topic | Geometric Topology Symplectic Geometry 57K33 (Primary) 57K10, 53D10 (Secondary) |
| url | https://arxiv.org/abs/2604.08010 |