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

 

Arc distance equals level number


Authors: Sangbum Cho, Darryl McCullough and Arim Seo
Journal: Proc. Amer. Math. Soc. 137 (2009), 2801-2807
MSC (2000): Primary 57M25
Published electronically: March 18, 2009
MathSciNet review: 2497495
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Abstract: Let $ K$ be a knot in $ 1$-bridge position with respect to a genus-$ g$ Heegaard surface that splits a $ 3$-manifold $ M$ into two handlebodies $ V$ and $ W$. One can move $ K$ by isotopy keeping $ K\cap V$ in $ V$ and $ K\cap W$ in $ W$ so that $ K$ lies in a union of $ n$ parallel genus-$ g$ surfaces tubed together by $ n-1$ straight tubes, and $ K$ intersects each tube in two arcs connecting the ends. We prove that the minimum $ n$ for which this is possible is equal to a Hempel-type distance invariant defined using the arc complex of the two-holed genus-$ g$ surface.


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Additional Information

Sangbum Cho
Affiliation: Department of Mathematics, University of California, Riverside, California 92521
Email: scho@math.ucr.edu

Darryl McCullough
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
Email: dmccullough@math.ou.edu

Arim Seo
Affiliation: Department of Mathematics, California State University, San Bernardino, California 92407
Email: aseo@csusb.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09874-8
PII: S 0002-9939(09)09874-8
Received by editor(s): September 22, 2008
Received by editor(s) in revised form: January 7, 2009
Published electronically: March 18, 2009
Additional Notes: The second author was supported in part by NSF grant DMS-0802424
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.