Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Distance between toroidal surgeries on hyperbolic knots in the $3$-sphere

Author: Masakazu Teragaito
Journal: Trans. Amer. Math. Soc. 358 (2006), 1051-1075
MSC (2000): Primary 57M25
Published electronically: April 13, 2005
Erratum: Trans. Amer. Math. Soc. 361 (2009), 3373-3374.
Correction: Trans. Amer. Math. Soc. 272 (1982), 803-807.
MathSciNet review: 2187645
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For a hyperbolic knot in the $3$-sphere, at most finitely many Dehn surgeries yield non-hyperbolic $3$-manifolds. As a typical case of such an exceptional surgery, a toroidal surgery is one that yields a closed $3$-manifold containing an incompressible torus. The slope corresponding to a toroidal surgery, called a toroidal slope, is known to be integral or half-integral. We show that the distance between two integral toroidal slopes for a hyperbolic knot, except the figure-eight knot, is at most four.

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

  • 1. S. Boyer and X. Zhang, Reducing Dehn fillings and toroidal Dehn fillings, Topology Appl. 68 (1996), 285-303. MR 1377050 (97f:57018)
  • 2. M. Brittenham and Y. Q. Wu, The classification of exceptional surgeries on 2-bridge knots, Comm. Anal. Geom. 9 (2001), 97-113. MR 1807953 (2001m:57008)
  • 3. M. Culler, C. McA. Gordon, J. Luecke and P. Shalen, Dehn surgery on knots, Ann. of Math. 125 (1987), 237-300. MR 0881270 (88a:57026)
  • 4. M. Eudave-Muñoz, Non-hyperbolic manifolds obtained by Dehn surgery on hyperbolic knots, Geometric topology (Athens, GA, 1993), 35-61, AMS/IP Stud. Adv. Math., 2.1, Amer. Math. Soc., Providence, RI, 1997. MR 1470720 (98i:57007)
  • 5. M. Eudave-Muñoz, On hyperbolic knots with Seifert fibered Dehn surgeries, Topology Appl. 121 (2002), 119-141. MR 1903687 (2003c:57005)
  • 6. D. Gabai, Foliations and the topology of 3-manifolds, III, J. Differential Geom. 26 (1987), 479-536. MR 0910018 (89a:57014b)
  • 7. C. McA. Gordon, Boundary slopes of punctured tori in 3-manifolds, Trans. Amer. Math. Soc. 350 (1998), 1713-1790. MR 1390037 (98h:57032)
  • 8. C. McA. Gordon and J. Luecke, Dehn surgeries on knots creating essential tori, I, Comm. Anal. Geom. 3 (1995), 597-644. MR 1371211 (96k:57003)
  • 9. C. McA. Gordon and J. Luecke, Dehn surgeries on knots creating essential tori. II, Comm. Anal. Geom. 8 (2000), 671-725. MR 1792371 (2002b:57003)
  • 10. C. McA. Gordon and J. Luecke, Non-integral toroidal Dehn surgeries, preprint.
  • 11. C. McA. Gordon and J. Luecke, Toroidal and boundary-reducing Dehn fillings, Topology Appl. 93 (1999), 77-90. MR 1684214 (2000b:57030)
  • 12. C. McA. Gordon and Y. Q. Wu, Toroidal and annular Dehn fillings, Proc. London Math. Soc. 78 (1999), 662-700. MR 1674841 (2000b:57029)
  • 13. C. McA. Gordon, Y. Q. Wu and X. Zhang, Non-integral toroidal surgery on hyperbolic knots in $S^3$, Proc. Amer. Math. Soc. 128 (2000), 1869-1879. MR 1644022 (2000j:57012)
  • 14. R. Kirby, Problems in low-dimensional topology, AMS/IP Stud. Adv. Math., 2.2, Geometric topology (Athens, GA, 1993), 35-473, Amer. Math. Soc., Providence, RI, 1997. MR 1470751
  • 15. S. Oh, Reducible and toroidal 3-manifolds obtained by Dehn fillings, Topology Appl. 75 (1997), 93-104. MR 1425387 (98a:57027)
  • 16. D. Rolfsen, Knots and links, Mathematics Lecture Series, 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288 (58:24236)
  • 17. M. Teragaito, Creating Klein bottles by surgery on knots, J. Knot Theory Ramifications 10 (2001), 781-794. MR 1839702 (2002f:57017)
  • 18. M. Teragaito, Toroidal surgeries on hyperbolic knots, II, Asian J. Math. 7 (2003), 139-146. MR 2015246
  • 19. W. Thurston, The geometry and topology of 3-manifolds, Princeton University, 1978.
  • 20. Y. Q. Wu, Dehn fillings producing reducible manifolds and toroidal manifolds, Topology 37 (1998), 95-108. MR 1480879 (98j:57033)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57M25

Retrieve articles in all journals with MSC (2000): 57M25

Additional Information

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

Keywords: Dehn surgery, toroidal surgery, knot
Received by editor(s): December 10, 2003
Received by editor(s) in revised form: April 7, 2004
Published electronically: April 13, 2005
Additional Notes: This work was partially supported by the Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), 14540082.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society