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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Online-Zugang: | https://arxiv.org/abs/2312.05014 |
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| _version_ | 1866910069753380864 |
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| author | Belousov, Yury |
| author_facet | Belousov, Yury |
| contents | It is known that each knot has a semimeander diagram (i. e. a diagram composed of two smooth simple arcs), however the number of crossings in such a diagram can only be roughly estimated. In the present paper we provide a new estimate of the complexity of the semimeander diagrams. We prove that for each knot $K$ with more than 10 crossings, there exists a semimeander diagram with no more than $0.31 \cdot 1.558^{\operatorname{cr}(K)}$ crossings, where $\operatorname{cr}(K)$ is the crossing number of $K$. As a corollary, we provide new estimates of the complexity of other meander-like types of knot diagrams, such as meander diagrams and potholders. We also describe an efficient algorithm for constructing a semimeander diagram from a given one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_05014 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the complexity of meander-like diagrams of knots Belousov, Yury Geometric Topology 57K10 It is known that each knot has a semimeander diagram (i. e. a diagram composed of two smooth simple arcs), however the number of crossings in such a diagram can only be roughly estimated. In the present paper we provide a new estimate of the complexity of the semimeander diagrams. We prove that for each knot $K$ with more than 10 crossings, there exists a semimeander diagram with no more than $0.31 \cdot 1.558^{\operatorname{cr}(K)}$ crossings, where $\operatorname{cr}(K)$ is the crossing number of $K$. As a corollary, we provide new estimates of the complexity of other meander-like types of knot diagrams, such as meander diagrams and potholders. We also describe an efficient algorithm for constructing a semimeander diagram from a given one. |
| title | On the complexity of meander-like diagrams of knots |
| topic | Geometric Topology 57K10 |
| url | https://arxiv.org/abs/2312.05014 |