Divides with cusps, shadows, and transvergent diagrams
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911650482749440 |
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| author | Furutani, Ryoga |
| author_facet | Furutani, Ryoga |
| contents | A link $L$ in $S^3$ is called a symmetric link if it is preserved by a $π$ rotation around a closed geodesic in $S^3$. Any symmetric link can be depicted by a diagram with a symmetry axis lying on the plane of the diagram, called a transvergent diagram. Recently, Sugawara proved that any symmetric link can be represented by a divide with cusps, which is a generalization of A'Campo's divide that allows a finite number of cusps. In this paper, we introduce a generalization of A'Campo's divide in terms of Turaev's shadow, called a divide with gleams. By using divides with gleams, we provide an algorithm to obtain a divide with cusps that represents a symmetric link from its given transvergent diagram. Conversely, we also provide an algorithm to draw a transvergent diagram of the link of a given divide with cusps. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_02385 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Divides with cusps, shadows, and transvergent diagrams Furutani, Ryoga Geometric Topology 57K10, 57Q60, 57R05 A link $L$ in $S^3$ is called a symmetric link if it is preserved by a $π$ rotation around a closed geodesic in $S^3$. Any symmetric link can be depicted by a diagram with a symmetry axis lying on the plane of the diagram, called a transvergent diagram. Recently, Sugawara proved that any symmetric link can be represented by a divide with cusps, which is a generalization of A'Campo's divide that allows a finite number of cusps. In this paper, we introduce a generalization of A'Campo's divide in terms of Turaev's shadow, called a divide with gleams. By using divides with gleams, we provide an algorithm to obtain a divide with cusps that represents a symmetric link from its given transvergent diagram. Conversely, we also provide an algorithm to draw a transvergent diagram of the link of a given divide with cusps. |
| title | Divides with cusps, shadows, and transvergent diagrams |
| topic | Geometric Topology 57K10, 57Q60, 57R05 |
| url | https://arxiv.org/abs/2503.02385 |