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

Authors:
Erica Flapan and Hugh Howards

Journal:
Proc. Amer. Math. Soc. **137** (2009), 4275-4285

MSC (2000):
Primary 57M25; Secondary 05C10

Published electronically:
July 20, 2009

MathSciNet review:
2538588

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Abstract: In contrast with knots, whose properties depend only on their extrinsic topology in , there is a rich interplay between the intrinsic structure of a graph and the extrinsic topology of all embeddings of the graph in . For example, it was shown by Conway and Gordon that every embedding of the complete graph in contains a non-trivial knot. Later it was shown that for every there is a complete graph such that every embedding of in contains a knot whose minimal crossing number is at least . Thus there are arbitrarily complicated knots in every embedding of a sufficiently large complete graph in . We prove the contrasting result that every graph has an embedding in such that every non-trivial knot in that embedding is hyperbolic. Our theorem implies that every graph has an embedding in which contains no composite or satellite knots.

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Additional Information

**Erica Flapan**

Affiliation:
Department of Mathematics, Pomona College, 610 North College Avenue, Claremont, California 91711-6348

**Hugh Howards**

Affiliation:
Department of Mathematics, Wake Forest University, P.O. Box 7388, Winston-Salem, North Carolina 27109-7388

DOI:
https://doi.org/10.1090/S0002-9939-09-09972-9

Received by editor(s):
October 31, 2008

Received by editor(s) in revised form:
March 16, 2009

Published electronically:
July 20, 2009

Communicated by:
Alexander N. Dranishnikov

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.