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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.05014 |
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Table of 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.