#### Document Type

Article

#### Department

Mathematics (Pomona)

#### Publication Date

2009

#### Keywords

knots, non-hyperbolic knot

#### Abstract

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.

#### Rights Information

© 2009 American Mathematical Society

#### Terms of Use & License Information

#### DOI

10.1090/S0002-9939-09-09972-9

#### Recommended Citation

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