knots, 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 K_7 in S³ contains a non-trivial knot. Later it was shown that for every m ∈ N there is a complete graph K_n such that every embedding of K_n 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.
© 2009 American Mathematical Society
E. Flapan and H. Howards, Every graph has an embedding in S3 containing no non-hyperbolic knot, Proceedings of the AMS, Vol 137 (2009) 4275-4285. doi: 10.1090/S0002-9939-09-09972-9