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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Arc distance equals level number
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by Sangbum Cho, Darryl McCullough and Arim Seo PDF
Proc. Amer. Math. Soc. 137 (2009), 2801-2807 Request permission

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
  • © 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), 2801-2807
  • MSC (2000): Primary 57M25
  • DOI: https://doi.org/10.1090/S0002-9939-09-09874-8
  • MathSciNet review: 2497495