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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Every graph has an embedding in $ S^3$ 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 $ 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$. For example, it was shown by Conway and Gordon that every embedding of the complete graph $ K_7$ in $ S^3$ contains a non-trivial knot. Later it was shown that for every $ m\in N$ there is a complete graph $ K_n$ such that every embedding of $ K_n$ in $ S_3$ 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 $ S^3$. We prove the contrasting result that every graph has an embedding in $ 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$ 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: http://dx.doi.org/10.1090/S0002-9939-09-09972-9
PII: S 0002-9939(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.