Abstract:Let $K$ be the graph on vertices $\{1, 2, 3, 4, 5\}$, and having all edges except $(4, 5)$. A continuous map $f:K\to \R^2$ is called an \emph{almost embedding} if $f$-images of non-adjacent edges are disjoint. Take the winding numbers of the $f$-image of the oriented cycle $(1, 2, 3)$ around $f(4)$ and around $f(5)$. We prove that the difference of these numbers equals $\pm 1$. This is surprising, because in other similar situations analogous statement is wrong.
From: Timur Garaev [view email]
[v1]
Sat, 25 Mar 2023 15:49:13 UTC (1,441 KB)
[v2]
Sat, 11 May 2024 10:20:43 UTC (699 KB)
[v3]
Sun, 1 Jun 2025 14:35:35 UTC (306 KB)
[v4]
Fri, 7 Aug 2026 18:31:06 UTC (339 KB)
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