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.References
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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