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

Emil Alkin, Alexander Miroshnikov

TL;DR

This work studies almost embeddings of the complete graph on four vertices, $K_4$, in the plane and assigns to each vertex $j$ a winding number $w_f(j)$ around the corresponding cycle $C_j$. The authors prove that the only constraint among the four integers is their odd total, and that for any triple of integers with odd sum there exists an almost embedding realizing those values. The approach is constructive, combining simple almost embeddings with a finger-move technique to adjust winding numbers, and is organized around two key lemmas that yield the desired realizations. The results connect to Radon and van Kampen-type invariants in planar topology and provide explicit geometric constructions that illuminate the landscape of planar almost embeddings for $K_4$.

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.

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

TL;DR

This work studies almost embeddings of the complete graph on four vertices, , in the plane and assigns to each vertex a winding number around the corresponding cycle . The authors prove that the only constraint among the four integers is their odd total, and that for any triple of integers with odd sum there exists an almost embedding realizing those values. The approach is constructive, combining simple almost embeddings with a finger-move technique to adjust winding numbers, and is organized around two key lemmas that yield the desired realizations. The results connect to Radon and van Kampen-type invariants in planar topology and provide explicit geometric constructions that illuminate the landscape of planar almost embeddings for .

Abstract

Let be the complete graph on four vertices. Let be a continuous map of to the plane such that -images of non-adjacent edges are disjoint. For any vertex take the winding number of the -image of the cycle around . 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.
Paper Structure (8 sections, 7 theorems, 36 equations, 4 figures)

This paper contains 8 sections, 7 theorems, 36 equations, 4 figures.

Table of Contents

  1. Introduction and the main result
  2. Some motivation
  3. Basic definitions
  4. A reduction of Theorem \ref{['t:main']}
  5. Winding number of non-closed polygonal line and its properties
  6. Proof of Lemma \ref{['l:str_al_em']}
  7. Proof of Lemma \ref{['l:n_finger_moves']}
  8. Addendum: more explicit proof of Lemma \ref{['l:construction']}

Key Result

Theorem 1.1

For any continuous almost embedding $f : K_4 \to {\mathbb R}^2$ we have $\sum_{j=1}^4 w_f(j) \equiv 1 \pmod 2$.

Figures (4)

  • Figure 1: Polygonal lines $A \ldots O$ (red line) and $B \ldots C$ (blue line) for $m = 1$ (left) and for $m = -1$ (right). 'Spiral' parts of both polygonal lines are depicted as continious lines. Dotted lines refer to the border $\partial R$ for $\varepsilon_1 = \frac{|AO|}{3}$ and $\varepsilon_2 = \frac{2|AO|}{3}$.
  • Figure 2: 'Finger moves' of a polygonal line $f|_\tau$ around a segment $f(\sigma)$: positive (left) and negative (right)
  • Figure 3: Left: images of edges $13$ and $24$ under the map $g$. Middle: $L$ is a simple closed (black) line disjoint from $g|_{13}$ and $g|_{24}$; $P$ is a (green) line disjoint from $g|_{24}$ and joining the point $g(3)$ to a vertex of $L$. Right: images of edges $13$ and $24$ under the map $f$.
  • Figure 4: Polygonal lines $A \ldots O$ (red line) and $B \ldots C$ (blue line) for $m = 2$

Theorems & Definitions (10)

  • Theorem 1.1: folklore
  • Theorem 1.2
  • Theorem 2.1: folklore
  • Theorem 2.2: Ga23
  • Remark 2.3: relation to the Radon and van Kampen numbers
  • Lemma 4.1
  • Lemma 4.2
  • proof : Deduction of Theorem \ref{['t:main']} from Lemmas \ref{['l:str_al_em']} and \ref{['l:n_finger_moves']}
  • Lemma 6.1
  • proof : Proof of Lemma \ref{['l:construction']}