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Proceedings of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9939-09-09874-8
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
MR Author ID: 830719
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

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.