Mathematische Zeitschrift

, Volume 267, Issue 3–4, pp 781–802

Lens spaces and toroidal Dehn fillings



We show that if M is a hyperbolic 3-manifold with ∂M a torus such that M(r1) is a lens space and M(r2) is toroidal, then Δ(r1, r2) ≤ 4.


Toroidal manifolds Lens spaces Dehn fillings 

Mathematics Subject Classification (2000)

Primary 57M50 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Goda H., Teragaito M.: Dehn surgeries on knots which yield lens spaces and genera of knots. Math. Proc. Camb. Philos. Soc 129, 501–515 (2000)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Goda H., Teragaito M.: On hyperbolic 3-manifolds realizing the maximal distance between toroidal Dehn fillings. Algebr. Geom. Topol. 5, 463–507 (2005)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Gordon C.McA.: Boundary slopes on punctured tori in 3-manifolds. Trans. Am. Math. Soc. 350, 1713–1790 (1998)MATHCrossRefGoogle Scholar
  4. 4.
    Gordon C.McA.: Toroidal Dehn surgeries on knots in lens spaces. Math. Proc. Camb. Philos. Soc. 125, 433–440 (1999)MATHCrossRefGoogle Scholar
  5. 5.
    Gordon, C.McA.: Small surfaces and Dehn fillings. In: Proceedings of the Kirbyfest, Geometry and Topology Monographs, vol. 2, pp. 177–199 (1999) (electronic)Google Scholar
  6. 6.
    Gordon, C.McA.: Combinatorial methods in Dehn surgery, Lectures at KNOTS ’96 (Tokyo), 263–290, Ser. Knots Everything, vol. 15. World Scientific Publication, River Edge, NJ (1997)Google Scholar
  7. 7.
    Gordon C.McA., Litherland R.A.: Incompressible planar surfaces in 3-manifolds. Topol. Appl. 18, 121–144 (1984)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Gordon C.McA., Luecke J.: Dehn surgeries on knots creating essential tori, I. Commun. Anal. Geom. 3, 597–644 (1995)MathSciNetMATHGoogle Scholar
  9. 9.
    Gordon C.McA., Luecke J.: Toroidal and boundary-reducing Dehn fillings. Topol. Appl. 93, 77–90 (1999)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Gordon C.McA., Wu Y.-Q.: Toroidal and annular Dehn fillings. Proc. Lond. Math. Soc 78, 662–700 (1999)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Hayashi C., Motegi K.: Only single twists on unknots can produce composite knots. Trans. Am. Math. Soc. 349, 4465–4479 (1997)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Jin G., Lee S., Oh S., Teragaito M.: P 2-reducing and toroidal Dehn fillings. Math. Proc. Camb. Philos. Soc 134, 271–288 (2003)MathSciNetMATHGoogle Scholar
  13. 13.
    Lee S.: Toroidal Dehn surgeries on konts in S 1 × S 2. J. Knot Theory Ramif. 14, 657–664 (2005)MATHCrossRefGoogle Scholar
  14. 14.
    Lee, S.: Dehn fillings yielding Klein bottles, International Mathematics Research Notices, vol. 2006, 34 pp. (2006). Article ID 24253Google Scholar
  15. 15.
    Lee S.: Exceptional Dehn fillings on hyperbolic 3-manifolds with at least two boundary components. Topology 46, 437–468 (2007)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Lee, S.: Klein bottle and toroidal Dehn fillings at distance 5 (preprint)Google Scholar
  17. 17.
    Lee S., Teragaito M.: Boundary structures of hyperbolic 3-manifolds admitting annular and toroidal fillings at large distance. Can. J. Math. 60, 164–188 (2008)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Teragaito M.: Distances between toroidal Dehn surgeries on hyperbolic knots in the 3-sphere. Trans. Am. Math. Soc 358, 1051–1075 (2006)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Teragaito M.: Toroidal Dehn fillings on large hyperbolic 3-manifolds. Commun. Anal. Geom. 14, 565–601 (2006)MathSciNetMATHGoogle Scholar
  20. 20.
    Teragaito, M.: Toroidal Dehn fillings and lens spaces containing Klein bottles, unpublished manuscriptGoogle Scholar
  21. 21.
    Valdez-Śanchez L.: Toroidal and Klein bottle boundary slopes. Topol. Appl. 154, 584–603 (2007)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsChung-Ang UniversitySeoulKorea

Personalised recommendations