Siberian Mathematical Journal

, Volume 44, Issue 4, pp 645–658 | Cite as

Spontaneous Surgery on the Borromean Rings

  • M. G. Pashkevich


We study the cone-manifolds whose singular sets are obtained by orbifold and spontaneous surgeries on components of the Borromean rings. We establish existence of geometric structures on these manifolds. For manifolds with hyperbolic structure we obtain an integral representation for volumes.

Dehn surgery link cone-manifold Borromean rings 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Hilden H. M., Lozano M. T., and Montesinos-Amilibia J. M., “On a remarkable polyhedron geometrizing the figure eight knot cone manifolds,” J. Math. Sci., Tokyo 2, 3, 501–561 (1995).Google Scholar
  2. 2.
    Kellerhals R., “On the volume of hyperbolic polyhedra,” Math. Ann., 285, 541–569 (1989).Google Scholar
  3. 3.
    Derevnin D. A. and Mednykh A. D., On the Volume of Spherical Lambert Cube, Research Institute of Mathematics Global Analysis Research Center, Seoul National University (2003). (Preprint Series 03-02).Google Scholar
  4. 4.
    Rivin I. and Hodgson C. D., “A characterization of compact convex polyhedra in hyperbolic 3-space,” Invent. Math., 111, No. 1, 77–111 (1993). (Corrigendum: 117, No. 2, 359 (1994).)Google Scholar
  5. 5.
    Hilden H. M., Lozano M. T., and Montesinos-Amilibia J. M., “On the Borromean orbifolds: geometry and arithmetic,” in: Topology'90, de Gruyter, Berlin, 1992, 1, pp. 133–167.Google Scholar
  6. 6.
    Thurston W., The Geometry and Topology of Three-Manifolds, Princeton Univ. Press, Princeton (1980).Google Scholar
  7. 7.
    Diaz R., “A characterization of Gram matrices of polytopes,” Discrete Comput. Geom., 21, 581–601 (1999).Google Scholar
  8. 8.
    Vinberg È. B., Geometry-2. Contemporary Problems of Mathematics. Vol. 29 [in Russian], VINITI, Moscow (1988). (Itogi Nauki i Tekhniki)Google Scholar
  9. 9.
    Berdon A. F., The Geometry of Discrete Groups [Russian translation], Nauka, Moscow (1986).Google Scholar
  10. 10.
    Weeks J., “SnapPea.” Software for hyperbolic 3-manifolds, available at Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • M. G. Pashkevich
    • 1
  1. 1.Sobolev Institute of MathematicsNovosibirsk

Personalised recommendations