On winding numbers of almost embeddings of $K_4$ in the plane

Emil Alkin; Alexander Miroshnikov

doi:10.1134/s0001434625603338

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arXiv · 2501.15642

On winding numbers of almost embeddings of $K_4$ in the plane

Emil Alkin·Alexander Miroshnikov

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Abstract

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.

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BibTeX RIS

Emil Alkin, Alexander Miroshnikov. 2025-01-26. On winding numbers of almost embeddings of $K_4$ in the plane. https://doi.org/10.1134/s0001434625603338

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