Every graph has an embedding in 𝑆³ containing no non-hyperbolic knot

Abstract

In contrast with knots, whose properties depend only on their extrinsic topology in S 3 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 S^3 . For example, it was shown by Conway and Gordon that every embedding of the complete graph K 7 K_7 in S 3 S^3 contains a non-trivial knot. Later it was shown that for every m ∈ N m\in N there is a complete graph K n K_n such that every embedding of K n K_n in S 3 S_3 contains a knot Q Q whose minimal crossing number is at least m m . Thus there are arbitrarily complicated knots in every embedding of a sufficiently large complete graph in S 3 S^3 . We prove the contrasting result that every graph has an embedding in S 3 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 S^3 which contains no composite or satellite knots.