Minimum numbers of Dehn colors of knots and $\mathcal{R}$-palette graphs
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910904885444608 |
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| author | Matsudo, Eri Oshiro, Kanako Yamagishi, Gaishi |
| author_facet | Matsudo, Eri Oshiro, Kanako Yamagishi, Gaishi |
| contents | In this paper, we consider minimum numbers of colors of knots for Dehn colorings. In particular, we will show that for any odd prime number $p$ and any Dehn $p$-colorable knot $K$, the minimum number of colors for $K$ is at least $\lfloor \log_2 p \rfloor +2$. Moreover, we will define the $\R$-palette graph for a set of colors. The $\R$-palette graphs are quite useful to give candidates of sets of colors which might realize a nontrivially Dehn $p$-colored diagram. In Appendix, we also prove that for Dehn $5$-colorable knot, the minimum number of colors is $4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_09941 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minimum numbers of Dehn colors of knots and $\mathcal{R}$-palette graphs Matsudo, Eri Oshiro, Kanako Yamagishi, Gaishi Geometric Topology In this paper, we consider minimum numbers of colors of knots for Dehn colorings. In particular, we will show that for any odd prime number $p$ and any Dehn $p$-colorable knot $K$, the minimum number of colors for $K$ is at least $\lfloor \log_2 p \rfloor +2$. Moreover, we will define the $\R$-palette graph for a set of colors. The $\R$-palette graphs are quite useful to give candidates of sets of colors which might realize a nontrivially Dehn $p$-colored diagram. In Appendix, we also prove that for Dehn $5$-colorable knot, the minimum number of colors is $4$. |
| title | Minimum numbers of Dehn colors of knots and $\mathcal{R}$-palette graphs |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2501.09941 |