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

 

Thin position and essential planar surfaces


Author: Ying-Qing Wu
Journal: Proc. Amer. Math. Soc. 132 (2004), 3417-3421
MSC (2000): Primary 57M25
Published electronically: June 17, 2004
MathSciNet review: 2073319
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Abstract: Abby Thompson proved that if a link $K$ is in thin position but not in bridge position, then the knot complement contains an essential meridional planar surface, and she asked whether some thin level surface must be essential. This note is to give a positive answer to this question, showing that if a link is in thin position but not bridge position, then a thinnest level surface is essential. A theorem of Rieck and Sedgwick follows as a consequence, which says that thin position of a connected sum of small knots comes in the obvious way.


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Additional Information

Ying-Qing Wu
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
Email: wu@math.uiowa.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07416-7
PII: S 0002-9939(04)07416-7
Keywords: Thin position, knots and links, essential planar surfaces
Received by editor(s): February 27, 2003
Received by editor(s) in revised form: June 16, 2003
Published electronically: June 17, 2004
Additional Notes: Partially supported by NSF grant #DMS 0203394
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2004 American Mathematical Society