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Toroidal surgeries on hyperbolic knots

Author: Masakazu Teragaito
Journal: Proc. Amer. Math. Soc. 130 (2002), 2803-2808
MSC (2000): Primary 57M50
Published electronically: February 4, 2002
MathSciNet review: 1900888
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Abstract: For a hyperbolic knot $K$ in $S^3$, a toroidal surgery is Dehn surgery which yields a $3$-manifold containing an incompressible torus. It is known that a toroidal surgery on $K$ is an integer or a half-integer. In this paper, we prove that all integers occur among the toroidal slopes of hyperbolic knots with bridge index at most three and tunnel number one.

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  • 1. S. Boyer and X. Zhang, Cyclic surgery and boundary slopes, in Geometric topology (Athens, GA, 1993), 62-79, AMS/IP Stud. Adv. Math., 2.1, Amer. Math. Soc., Providence, RI, 1997. MR 98i:57005
  • 2. M. Brittenham and Y.Q. Wu. The classification of exceptional surgeries on 2-bridge knots, Comm. Anal. Geom. 9 (2001), 97-113. CMP 2001:07
  • 3. A. Casson and C. McA. Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987), 275-283. MR 89c:57020
  • 4. J. Dean, Hyperbolic knots with small Seifert-fibered Dehn surgeries, Ph.D. Thesis, The University of Texas at Austin, 1996.
  • 5. M. Eudave-Muñoz, Non-hyperbolic manifolds obtained by Dehn surgery on hyperbolic knots, in Geometric topology (Athens, GA, 1993), 35-61, AMS/IP Stud. Adv. Math., 2.1, Amer. Math. Soc., Providence, RI, 1997. MR 98i:57007
  • 6. M. Eudave-Muñoz, On hyperbolic knots with Seifert fibered Dehn surgeries, preprint.
  • 7. C. McA. Gordon, Dehn filling: a survey, Knot theory (Warsaw, 1995), 129-144, Banach Center Publ., 42, Polish Acad. Sci., Warsaw, 1998. MR 99e:57028
  • 8. C. McA. Gordon and J. Luecke, Dehn surgeries on knots creating essential tori, I, Communications in Analysis and Geometry 3 (1995), 597-644. MR 96k:57003
  • 9. L. Moser, Elementary surgery along a torus knot, Pacific J. Math. 38 (1971), 737-745. MR 52:4287
  • 10. R. Patton, Incompressible punctured tori in the complements of alternating knots, Math. Ann. 301 (1995), 1-22. MR 95k:57011
  • 11. D. Rolfsen, Knots and links, Mathematics Lecture Series, 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 95c:57018
  • 12. W. Thurston, The geometry and topology of 3-manifolds, Princeton University, 1978.
  • 13. F. Waldhausen, Gruppen mit Zentrum und 3-dimensionale Mannigfaltigkeiten, Topology 6 (1967), 505-517. MR 38:5223

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Additional Information

Masakazu Teragaito
Affiliation: Department of Mathematics and Mathematics Education, Faculty of Education, Hiroshima University, 1-1-1 Kagamiyama, Higashi-hiroshima 739-8524, Japan

Received by editor(s): December 6, 2000
Received by editor(s) in revised form: April 18, 2001
Published electronically: February 4, 2002
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2002 American Mathematical Society

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