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

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.