Every graph has an embedding in $S^3$ containing no non-hyperbolic knot

TL;DR

This paper proves that every graph can be embedded in the 3-sphere in such a way that all non-trivial knots in the embedding are hyperbolic, contrasting with previous results about unavoidable non-trivial knots.

Contribution

It introduces a method to embed any graph in $S^3$ avoiding non-hyperbolic knots, showing a new topological property of graph embeddings.

Findings

Every graph has an embedding in $S^3$ with only hyperbolic knots.

Such embeddings contain no composite or satellite knots.

Contrasts with prior results where embeddings necessarily contain complex knots.

Abstract

In contrast with knots, whose properties depend only on their extrinsic topology in $S^3$, there is a rich interplay between the intrinsic structure of a graph and the extrinsic topology of all embeddings of the graph in $S^3$ . For example, it was shown in [2] that every embedding of the complete graph $K_7$ in $S^3$ contains a non-trivial knot. Later in it was shown that for every $m \in N$, there is a complete graph $K_n$ such that every embedding of $K_n$ in $S_3$ contains a knot $Q$ whose minimal crossing number is at least $m$. Thus there are arbitrarily complicated knots in every embedding of a sufficiently large complete graph in $S^3$. We prove here the contrasting result that every graph has an embedding in $S^3$ such that every non-trivial knot in that embedding is hyperbolic. Our theorem implies that every graph has an embedding in $S^3$ which contains no composite or satellite knots.