Every Graph Has an Embedding in S3 Containing No Non-hyperbolic Knot Erica Flapan and Hugh Howards

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 S 3 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 S 3 contains a knot Q whose minimal crossing number is at least m. 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 we can add local knots within every edge to get an embedding such that every knot in that embedding is composite. Let G be a graph. There is an odd number n, such that G is a minor of Kn. We will show that for every odd number n, there is an embedding of Kn in S 3 such that every non-trivial knot in that embedding of Kn is hyperbolic. It follows that there is an embedding of G in S3 which contains no non-trivial non-hyperbolic knots. Let n be a fixed odd number. We begin by constructing a preliminary embedding of Kn in S 3 as follows. Let h be a rotation of S3 of order n with fixed point set α ∼= S1. Let V denote the complement of an open regular neighborhood of the fixed point set α. Let v1, . . . , vn be points in V such that for each i, h(vi) = vi+1 (throughout the paper we shall consider