Every Graph Has an Embedding in S3 Containing No Non-hyperbolic Knot

In contrast with knots, whose properties depend only on their extrinsic topology in S3, there is a rich interplay between the intrinsic structure of a graph and the extrinsic topology of all embeddings of the graph in S3. For example, it was shown by Conway and Gordon that every embedding of the complete graph K7 in S3 contains a non-trivial knot. Later it was shown that for every m ∈ N there is a complete graph Kn such that every embedding of Kn in S3 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 S3. We prove the contrasting result that every graph has an embedding in S3 such that every non-trivial knot in that embedding is hyperbolic. Our theorem implies that every graph has an embedding in S3 which contains no composite or satellite knots. In contrast with knots, whose properties depend only on their extrinsic topology in S3, there is a rich interplay between the intrinsic structure of a graph and the extrinsic topology of all embeddings of the graph in S3. For example, it was shown in [2] that every embedding of the complete graph K7 in S3 contains a non-trivial knot. Later in [3] it was shown that for every m ∈ N, there is a complete graph Kn such that every embedding of Kn in S3 contains a knot Q (i.e., Q is a subgraph of Kn) such that |a2(Q)| ≥ m, where a2 is the second coefficient of the Conway polynomial of Q. More recently, in [4] it was shown that for every m ∈ N, there is a complete graph Kn such that every embedding of Kn in S3 contains a knot Q whose minimal crossing number is at leastm. Thus there are arbitrarily complicated knots (as measured by a2 and the minimal crossing number) in every embedding of a sufficiently large complete graph in S3. In light of these results, it is natural to ask whether there is a graph such that every embedding of that graph in S3 contains a composite knot. Or more generally, is there a graph such that every embedding of the graph in S3 contains a satellite knot? Certainly, K7 is not an example of such a graph since Conway and Gordon [2] exhibit an embedding of K7 containing only the trefoil knot. In this paper we answer this question in the negative. In particular, we prove that every graph has an embedding in S3 such that every non-trivial knot in that embedding is hyperbolic. Our theorem implies that every graph has an embedding in S3 which contains no composite or satellite knots. By contrast, for any particular embedding of a graph Received by the editors October 31, 2008, and, in revised form, March 16, 2009. 2000 Mathematics Subject Classification. Primary 57M25; Secondary 05C10. c ©2009 American Mathematical Society Reverts to public domain 28 years from publication