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

TL;DR

This paper provides an algebraic characterization of when disjoint embeddings of complete graphs in three-dimensional space are weakly linked, based on linking numbers and shared substructures.

Contribution

It introduces a novel algebraic criterion for weak linking of complete graphs in , expanding understanding of spatial graph embeddings.

Findings

Weakly linked pairs have specific shared vertex or triangle structures.

Characterization of all weakly linked pairs based on these structures.

Provides conditions for when complete graphs are weakly linked in .

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.