Intrinsically Linked Graphs with Knotted Components

TL;DR

This paper constructs graphs with guaranteed complex linked and knotted structures in three-dimensional space, revealing new topological properties of graph embeddings.

Contribution

It introduces graphs that necessarily contain knotted and linked components in any embedding into three-dimensional space, advancing topological graph theory.

Findings

Existence of graphs with knotted components in all embeddings

Construction of graphs with multiple nontrivial knots in links

Complete graphs contain links with nonzero multiple linking numbers

Abstract

We construct a graph G such that any embedding of G into R^{3} contains a nonsplit link of two components, where at least one of the components is a nontrivial knot. Further, for any m < n we produce a graph H so that every embedding of H contains a nonsplit n component link, where at least m of the components are nontrivial knots. We then turn our attention to complete graphs and show that for any given n, every embedding of a large enough complete graph contains a two component link whose linking number is a nonzero multiple of n.