Closure of knitted surfaces and surface-links
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917030900269056 |
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| author | Nakamura, Inasa Yasuda, Jumpei |
| author_facet | Nakamura, Inasa Yasuda, Jumpei |
| contents | A knitted surface is a surface with or without closed components smoothly properly embedded in $D^2 \times B^2$, which is a generalization of a braided surface. A knitted surface is called a 2-dimensional knit if its boundary is the closure of a trivial braid. From a 2-dimensional knit $S$, we obtain a surface-link in $\mathbb{R}^4$ by taking the closure of $S$. We show that any surface-link is ambient isotopic to the closure of some 2-dimensional knit. Further, we consider another type of the closure of a knitted surface, called the plat closure. It is known that any trivial surface-knot is ambient isotopic to the plat closure of a knitted surface of degree 2. We show that the plat closure of any knitted surface of degree 2 is a trivial surface-link, and any trivial surface-link is ambient isotopic to the plat closure of a knitted surface of degree $2$. We also show the same result for the closure of 2-dimensional knits of degree 2. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_19432 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Closure of knitted surfaces and surface-links Nakamura, Inasa Yasuda, Jumpei Geometric Topology A knitted surface is a surface with or without closed components smoothly properly embedded in $D^2 \times B^2$, which is a generalization of a braided surface. A knitted surface is called a 2-dimensional knit if its boundary is the closure of a trivial braid. From a 2-dimensional knit $S$, we obtain a surface-link in $\mathbb{R}^4$ by taking the closure of $S$. We show that any surface-link is ambient isotopic to the closure of some 2-dimensional knit. Further, we consider another type of the closure of a knitted surface, called the plat closure. It is known that any trivial surface-knot is ambient isotopic to the plat closure of a knitted surface of degree 2. We show that the plat closure of any knitted surface of degree 2 is a trivial surface-link, and any trivial surface-link is ambient isotopic to the plat closure of a knitted surface of degree $2$. We also show the same result for the closure of 2-dimensional knits of degree 2. |
| title | Closure of knitted surfaces and surface-links |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2412.19432 |