The linear minimal 4-chart with three crossings
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917449176186880 |
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| author | Nagase, Teruo Shima, Akiko |
| author_facet | Nagase, Teruo Shima, Akiko |
| contents | Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded surfaces in 4-space by using charts.
Let $Γ$ be a chart, and we denote by $Cross(Γ)$ the set of all the crossings of $Γ$, and we denote by $Γ_m$ the union of all the edges of label $m$. For a 4-chart $Γ$, if each connected component of the set $(Γ_1\cup Γ_3)-Cross(Γ)$ is acyclic, then $Γ$ is said to be {\it linear}. In this paper, we shall show that any linear minimal $4$-chart with three crossings is lor-equivalent (Label-Orientation-Reflection equivalent) to the chart describing a $2$-twist spun trefoil knot by omitting free edges and hoops. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_04114 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The linear minimal 4-chart with three crossings Nagase, Teruo Shima, Akiko Geometric Topology 57K45, 05C10, 57M15 Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded surfaces in 4-space by using charts. Let $Γ$ be a chart, and we denote by $Cross(Γ)$ the set of all the crossings of $Γ$, and we denote by $Γ_m$ the union of all the edges of label $m$. For a 4-chart $Γ$, if each connected component of the set $(Γ_1\cup Γ_3)-Cross(Γ)$ is acyclic, then $Γ$ is said to be {\it linear}. In this paper, we shall show that any linear minimal $4$-chart with three crossings is lor-equivalent (Label-Orientation-Reflection equivalent) to the chart describing a $2$-twist spun trefoil knot by omitting free edges and hoops. |
| title | The linear minimal 4-chart with three crossings |
| topic | Geometric Topology 57K45, 05C10, 57M15 |
| url | https://arxiv.org/abs/2509.04114 |