Minimal area of the spun trefoil knot on the canonical cubulation of $\mathbb{R}^4$
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911052836372480 |
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| author | Baray, Ana Catalán, Juan José Hinojosa, Gabriela Valdez, Rogelio |
| author_facet | Baray, Ana Catalán, Juan José Hinojosa, Gabriela Valdez, Rogelio |
| contents | We say that a \emph{cubical 2-knot} $K^{2}$ is an embedding of the 2-sphere in the 2-skeleton of the canonical cubulation of $\mathbb{R}^4$; in particular, $K^{2}$ is the union of $m(K^{2})$ unit squares, hence $m(K^{2})$ is its area. We define the minimal area of $K^{2}$ as the minimum over all the areas of cubical 2-knots isotopic to the given knot type. The minimal area of a cubical 2-knot is an invariant, and the following natural question arose: Given a knot type, what area is needed for a cubical 2-knot in the canonical cubulation of $\mathbb{R}^4$ to realise that type with minimal area? In this paper, we answer this question for the spun trefoil knot in the weakly minimal case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09361 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minimal area of the spun trefoil knot on the canonical cubulation of $\mathbb{R}^4$ Baray, Ana Catalán, Juan José Hinojosa, Gabriela Valdez, Rogelio Geometric Topology Primary 57Q15, Secondary 57Q35, 57Q05 We say that a \emph{cubical 2-knot} $K^{2}$ is an embedding of the 2-sphere in the 2-skeleton of the canonical cubulation of $\mathbb{R}^4$; in particular, $K^{2}$ is the union of $m(K^{2})$ unit squares, hence $m(K^{2})$ is its area. We define the minimal area of $K^{2}$ as the minimum over all the areas of cubical 2-knots isotopic to the given knot type. The minimal area of a cubical 2-knot is an invariant, and the following natural question arose: Given a knot type, what area is needed for a cubical 2-knot in the canonical cubulation of $\mathbb{R}^4$ to realise that type with minimal area? In this paper, we answer this question for the spun trefoil knot in the weakly minimal case. |
| title | Minimal area of the spun trefoil knot on the canonical cubulation of $\mathbb{R}^4$ |
| topic | Geometric Topology Primary 57Q15, Secondary 57Q35, 57Q05 |
| url | https://arxiv.org/abs/2507.09361 |