A note on Fox colorings of virtual tangles
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arXiv
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| Format: | Preprint |
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2026
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| author | Nakamura, Takuji Nakanishi, Yasutaka Satoh, Shin Wada, Kodai |
| author_facet | Nakamura, Takuji Nakanishi, Yasutaka Satoh, Shin Wada, Kodai |
| contents | We study Fox colorings of tangle diagrams by $R=\mathbb{Z}$ or $\mathbb{Z}/p\mathbb{Z}$, where $p\geq3$ is an odd integer. For an $R$-colored $m$-string tangle diagram, the colors at the $2m$ boundary points form a vector $v\in R^{2m}$. We show that for classical tangle diagrams, such vectors are completely characterized by the alternating sum condition $Δ(v)=0$. We then investigate how this restriction changes in the virtual setting. For $R=\mathbb{Z}$, the realizability of $v$ is determined by a divisibility condition on $Δ(v)$. For $R=\mathbb{Z}/p\mathbb{Z}$, every vector is realizable by a virtual tangle diagram. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_18392 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A note on Fox colorings of virtual tangles Nakamura, Takuji Nakanishi, Yasutaka Satoh, Shin Wada, Kodai Geometric Topology 57K10, 57K12 We study Fox colorings of tangle diagrams by $R=\mathbb{Z}$ or $\mathbb{Z}/p\mathbb{Z}$, where $p\geq3$ is an odd integer. For an $R$-colored $m$-string tangle diagram, the colors at the $2m$ boundary points form a vector $v\in R^{2m}$. We show that for classical tangle diagrams, such vectors are completely characterized by the alternating sum condition $Δ(v)=0$. We then investigate how this restriction changes in the virtual setting. For $R=\mathbb{Z}$, the realizability of $v$ is determined by a divisibility condition on $Δ(v)$. For $R=\mathbb{Z}/p\mathbb{Z}$, every vector is realizable by a virtual tangle diagram. |
| title | A note on Fox colorings of virtual tangles |
| topic | Geometric Topology 57K10, 57K12 |
| url | https://arxiv.org/abs/2603.18392 |