On winding numbers of almost embeddings of $K_4$ in the plane
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918010511425536 |
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| author | Alkin, Emil Miroshnikov, Alexander |
| author_facet | Alkin, Emil Miroshnikov, Alexander |
| contents | Let $K_4$ be the complete graph on four vertices. Let $f$ be a continuous map of $K_4$ to the plane such that $f$-images of non-adjacent edges are disjoint. For any vertex $v \in K_4$ take the winding number of the $f$-image of the cycle $K_4 - v$ around $f(v)$. It is known that the sum of these four integers is odd. We construct examples showing that this is the only relation between these four numbers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_15642 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On winding numbers of almost embeddings of $K_4$ in the plane Alkin, Emil Miroshnikov, Alexander Geometric Topology Computational Geometry Combinatorics 57M15, 55M25, 05C10 Let $K_4$ be the complete graph on four vertices. Let $f$ be a continuous map of $K_4$ to the plane such that $f$-images of non-adjacent edges are disjoint. For any vertex $v \in K_4$ take the winding number of the $f$-image of the cycle $K_4 - v$ around $f(v)$. It is known that the sum of these four integers is odd. We construct examples showing that this is the only relation between these four numbers. |
| title | On winding numbers of almost embeddings of $K_4$ in the plane |
| topic | Geometric Topology Computational Geometry Combinatorics 57M15, 55M25, 05C10 |
| url | https://arxiv.org/abs/2501.15642 |