Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Weakly linked embeddings of pairs of complete graphs in $\mathbb{R}^3$

James Di, Erica Flapan, Spencer Johnson, Daniel Thompson, Christopher Tuffley

Abstract

Let $G$ and $H$ be disjoint embeddings of complete graphs $K_m$ and $K_n$ in $\mathbb{R}^3$ such that some cycle in $G$ links a cycle in $H$ with non-zero linking number. We say that $G$ and $H$ are *weakly linked* if the absolute value of the linking number of any cycle in $G$ with a cycle in $H$ is $0$ or $1$. Our main result is an algebraic characterisation of when a pair of disjointly embedded complete graphs is weakly linked. As a step towards this result, we show that if $G$ and $H$ are weakly linked, then each contains either a vertex common to all triangles linking the other or a triangle which shares an edge with all triangles linking the other. All families of weakly linked pairs of complete graphs are then characterised by which of these two cases holds in each complete graph.

Weakly linked embeddings of pairs of complete graphs in $\mathbb{R}^3$

Abstract

Let and be disjoint embeddings of complete graphs and in such that some cycle in links a cycle in with non-zero linking number. We say that and are *weakly linked* if the absolute value of the linking number of any cycle in with a cycle in is or . Our main result is an algebraic characterisation of when a pair of disjointly embedded complete graphs is weakly linked. As a step towards this result, we show that if and are weakly linked, then each contains either a vertex common to all triangles linking the other or a triangle which shares an edge with all triangles linking the other. All families of weakly linked pairs of complete graphs are then characterised by which of these two cases holds in each complete graph.

Paper Structure

This paper contains 10 sections, 31 theorems, 68 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Weak linking of a simple closed curve with $K_n$
  3. Weak linking of a $\Theta$ curve with $K_n$
  4. Weak linking of $K_4$ with $K_n$
  5. Stars with no common apex
  6. Introduction
  7. Our main results
  8. Embeddings with a common triangle
  9. Embeddings with a common vertex in both $G$ and $H$
  10. Realising Theorem \ref{['thm:KmKn-pq']}

Key Result

Theorem 1.3

Let $m\geq 5$ and $n\geq 4$, and suppose that $G\cong K_m$ and $H\cong K_n$ are weakly linked in $\mathbb{R}^3$. Then exactly one of the following holds:

Figures (7)

  • Figure 1.1: An embedding of $K_6=\langle p,q_0,q_1,r_0,r_1,r_2\rangle$ and a curve $C$ such that $C$ links $K_6$ in the star $p|q_0q_1|r_0r_1r_2$.
  • Figure 2.1: Three triangles $p_0p_1q_2$, $p_0q_1p_2$, $q_0p_1p_2$ that pairwise intersect but share no common vertex.
  • Figure 3.1: Cases \ref{['case:commonapex']} (left) and \ref{['case:nocommonapex']} (right) of Theorem \ref{['thm:thetacurve']}. The second vertex of $\Theta$ is placed at infinity.
  • Figure 4.1: Embeddings of $G\cong K_4$ (blue) and $H\cong K_n$ (red) realising Cases \ref{['case:K4Kn-commonvertex']} (left) and \ref{['case:K4Kn-nocommonvertex']} (right) of Theorem \ref{['thm:K4Kn']}.
  • Figure 5.1: The cycles $C_1$, $C_2$ and $C_3$ in the proof of Proposition \ref{['prop:nocommonapex']}.
  • ...and 2 more figures

Theorems & Definitions (70)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: Theorem \ref{['thm:common-vertex-or-triangle']} paraphrased
  • Definition 1.4
  • Definition 1.5
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 60 more