Every graph has an embedding in $S^3$ containing no non-hyperbolic knot
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- by Erica Flapan and Hugh Howards
- Proc. Amer. Math. Soc. 137 (2009), 4275-4285
- DOI: https://doi.org/10.1090/S0002-9939-09-09972-9
- Published electronically: July 20, 2009
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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.References
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Bibliographic 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
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 4275-4285
- MSC (2000): Primary 57M25; Secondary 05C10
- DOI: https://doi.org/10.1090/S0002-9939-09-09972-9
- MathSciNet review: 2538588