On winding numbers of $K_5$ minus an edge in the plane

Abstract

Let $K$ be the graph on vertices $\{1,2,3,4,5\}$, and having all edges except $(4,5)$. A continuous map $f:K\to {\mathbb{R}}^2$ 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.