Linking number and folded ribbon unknots
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866915507047759872 |
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| author | Denne, Elizabeth Larsen, Troy |
| author_facet | Denne, Elizabeth Larsen, Troy |
| contents | We study Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The folded ribbonlength is the length to width ratio of such a folded ribbon knot. The folded ribbon knot is also a framed knot, and the ribbon linking number is the linking number of the knot and one boundary component of the ribbon. We find the minimum folded ribbonlength for $3$-stick unknots with ribbon linking numbers $\pm1$ and $\pm 3$, and we prove that the minimum folded ribbonlength for $n$-gons with obtuse interior angles is achieved when the $n$-gon is regular. Among other results, we prove that the minimum folded ribbonlength of any folded ribbon unknot which is a topological annulus with ribbon linking number $\pm n$ is bounded from above by $2n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_03239 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Linking number and folded ribbon unknots Denne, Elizabeth Larsen, Troy Geometric Topology 57K10 We study Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The folded ribbonlength is the length to width ratio of such a folded ribbon knot. The folded ribbon knot is also a framed knot, and the ribbon linking number is the linking number of the knot and one boundary component of the ribbon. We find the minimum folded ribbonlength for $3$-stick unknots with ribbon linking numbers $\pm1$ and $\pm 3$, and we prove that the minimum folded ribbonlength for $n$-gons with obtuse interior angles is achieved when the $n$-gon is regular. Among other results, we prove that the minimum folded ribbonlength of any folded ribbon unknot which is a topological annulus with ribbon linking number $\pm n$ is bounded from above by $2n$. |
| title | Linking number and folded ribbon unknots |
| topic | Geometric Topology 57K10 |
| url | https://arxiv.org/abs/2208.03239 |