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

TL;DR

This paper investigates the winding numbers associated with almost embeddings of the complete graph on four vertices in the plane, establishing that the sum of these numbers is always odd and that this is the only relation among them.

Contribution

The paper proves that the sum of the winding numbers around a vertex is always odd and constructs examples demonstrating this is the only relation among these numbers.

Findings

Sum of winding numbers is always odd.

This odd sum relation is unique among possible relations.

Constructed examples show no other constraints exist.

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.