Two-tone colorings and surjective dihedral representations for links
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866913331894288384 |
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| author | Ichihara, Kazuhiro Ishikawa, Katsumi Matsudo, Eri Suzuki, Masaaki |
| author_facet | Ichihara, Kazuhiro Ishikawa, Katsumi Matsudo, Eri Suzuki, Masaaki |
| contents | It is well-known that a knot is Fox $n$-colorable for a prime $n$ if and only if the knot group admits a surjective homomorphism to the dihedral group of degree $n$. However, this is not the case for links with two or more components. In this paper, we introduce a two-tone coloring on a link diagram, and give a condition for links so that the link groups admit surjective representations to the dihedral groups. In particular, it is shown that the link group of any link with at least 3 components admits a surjective homomorphism to the dihedral group of arbitrary degree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_13706 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Two-tone colorings and surjective dihedral representations for links Ichihara, Kazuhiro Ishikawa, Katsumi Matsudo, Eri Suzuki, Masaaki Geometric Topology 57K10 It is well-known that a knot is Fox $n$-colorable for a prime $n$ if and only if the knot group admits a surjective homomorphism to the dihedral group of degree $n$. However, this is not the case for links with two or more components. In this paper, we introduce a two-tone coloring on a link diagram, and give a condition for links so that the link groups admit surjective representations to the dihedral groups. In particular, it is shown that the link group of any link with at least 3 components admits a surjective homomorphism to the dihedral group of arbitrary degree. |
| title | Two-tone colorings and surjective dihedral representations for links |
| topic | Geometric Topology 57K10 |
| url | https://arxiv.org/abs/2302.13706 |