Constructing embedded surfaces for cellular embeddings of leveled spatial graphs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911217894817792 |
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| author | Barthel, Senja Buccoliero, Fabio |
| author_facet | Barthel, Senja Buccoliero, Fabio |
| contents | For a given spatial graph $\mathcal{G} \subset \mathbb{R}^3$, we would like to find a closed orientable surface $\mathcal{S}$ embedded in $\mathbb{R}^3$ in which $\mathcal{G}$ is cellular embedded. However, for general $\mathcal{G}$ this is not possible. We therefore define a property of spatial graphs, called leveled, to show that for leveled spatial graphs with a small number of levels, a surface $\mathcal{S}$ can always be found. The argument is based on decomposing $\mathcal{G}$ into spatial subgraphs that can be placed on a sphere and on cylinders attached as handles, in such a way that the resulting surface contains a cellular embedding of $\mathcal{G}$. We generalize the procedure to an algorithm that, if successful, constructs $\mathcal{S}$ for leveled spatial graphs with any number of levels. We conjecture that all connected leveled embeddings can be cellular embedded with the presented algorithm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_03800 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constructing embedded surfaces for cellular embeddings of leveled spatial graphs Barthel, Senja Buccoliero, Fabio Geometric Topology Combinatorics 57M15, 57M25, 05C10, 05C62, 05C45 For a given spatial graph $\mathcal{G} \subset \mathbb{R}^3$, we would like to find a closed orientable surface $\mathcal{S}$ embedded in $\mathbb{R}^3$ in which $\mathcal{G}$ is cellular embedded. However, for general $\mathcal{G}$ this is not possible. We therefore define a property of spatial graphs, called leveled, to show that for leveled spatial graphs with a small number of levels, a surface $\mathcal{S}$ can always be found. The argument is based on decomposing $\mathcal{G}$ into spatial subgraphs that can be placed on a sphere and on cylinders attached as handles, in such a way that the resulting surface contains a cellular embedding of $\mathcal{G}$. We generalize the procedure to an algorithm that, if successful, constructs $\mathcal{S}$ for leveled spatial graphs with any number of levels. We conjecture that all connected leveled embeddings can be cellular embedded with the presented algorithm. |
| title | Constructing embedded surfaces for cellular embeddings of leveled spatial graphs |
| topic | Geometric Topology Combinatorics 57M15, 57M25, 05C10, 05C62, 05C45 |
| url | https://arxiv.org/abs/2406.03800 |