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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
Published electronically: March 18, 2009
MathSciNet review: 2497495
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Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. D. Bachman, S. Schleimer, Distance and bridge position, Pacific J. Math. 219 (2005), 221-235. MR 2175113 (2007a:57028)
  • 2. S. Cho, Homeomorphisms of the $ 3$-sphere that preserve a Heegaard splitting of genus two, Proc. Amer. Math. Soc. 136 (2008), 1113-1123. MR 2361888
  • 3. M. Eudave-Muñoz, Incompressible surfaces in tunnel number one knot complements, II, Topology Appl. 98 (1999), 167-189. MR 1719999 (2000h:57010)
  • 4. M. Eudave-Muñoz, Incompressible surfaces and $ (1,1)$-knots, J. Knot Theory Ramifications 15 (2006), 935-948. MR 2251034 (2007g:57007)
  • 5. J. Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. (2) 121 (1985), 215-249. MR 786348 (87f:57009)
  • 6. J. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986), 157-176. MR 830043 (87c:32030)
  • 7. A. Hatcher, On triangulations of surfaces, Topology Appl. 40 (1991), 189-194. MR 1123262 (92f:57020)
  • 8. J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (2001), 631-657. MR 1838999 (2002f:57044)
  • 9. T. Kobayashi, Classification of unknotting tunnels for two bridge knots, in Proceedings of the Kirbyfest (Berkeley, CA, 1998), 259-290, Geom. Topol. Monogr. 2, Geom. Topol. Publ., Coventry, 1999. MR 1734412 (2000j:57013)
  • 10. K. Morimoto, M. Sakuma, On unknotting tunnels for knots, Math. Ann. 289 (1991), 143-167. MR 1087243 (92e:57015)
  • 11. R. C. Penner, The simplicial compactification of Riemann's moduli space, in Topology and Teichmüller spaces (Katinkulta, 1995), 237-252, World Sci. Publ., River Edge, NJ, 1996. MR 1659667 (99j:32023)

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Additional Information

Sangbum Cho
Affiliation: Department of Mathematics, University of California, Riverside, California 92521

Darryl McCullough
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019

Arim Seo
Affiliation: Department of Mathematics, California State University, San Bernardino, California 92407

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.

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