Siberian Mathematical Journal

, Volume 52, Issue 3, pp 537–543 | Cite as

Dehn surgeries on the figure eight knot: An upper bound for complexity



We establish an upper bound ω(p/q) on the complexity of the manifolds obtained by p/qsurgeries on the figure eight knot. It turns out that in case ω(p/q) ⩽ 12 the bound is sharp.


Dehn surgery figure eight knot complexity 


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  1. 1.
    Matveev S. V., “Complexity theory of three-dimensional manifolds,” Acta Appl. Math., 19, No. 2, 101–130 (1990).MATHMathSciNetGoogle Scholar
  2. 2.
    Frigerio R., Martelli B., and Petronio C., “Complexity and Heegaard genus of an infinite class of compact 3-manifolds,” Pacific J. Math., 210, No. 2, 283–297 (2003).MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Anisov S., “Exact values of complexity for an infinite number of 3-manifolds,” Moscow Math. J., 5, No. 2, 305–310 (2005).MATHMathSciNetGoogle Scholar
  4. 4.
    Jaco W., Rubinstein H., and Tillmann S., “Minimal triangulations for an infinite family of lens spaces,” J. Topology, 2, No. 1, 157–180 (2009).MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Matveev S. V., “Tabulation of three-dimensional manifolds,” Russian Math. Surveys, 60, No. 4, 673–698 (2005).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Fominykh E., Matveev S., and Tarkaev V., “Atlas of 3-manifolds,” available at
  7. 7.
    Matveev S. V., Algorithmic Topology and Classification of 3-Manifolds, Springer-Verlag, Berlin (2003) (Algorithms and Computation in Mathematics; V. 9).MATHGoogle Scholar
  8. 8.
    Martelli B. and Petronio C., “Complexity of geometric 3-manifolds,” Geom. Dedicata, 108, 15–69 (2004).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Fominykh E. and Ovchinnikov M., “On the complexity of graph-manifolds,” Siberian Electronic Math. Reports, 2, 190–191 (2005).MATHMathSciNetGoogle Scholar
  10. 10.
    Fominykh E. A., “Upper complexity bounds for an infinite family of graph-manifolds,” Siberian Electronic Math. Reports, 5, 215–228 (2008).MathSciNetGoogle Scholar
  11. 11.
    Rolfsen D., Knots and Links, Publish or Perish, Berkeley (1976).MATHGoogle Scholar
  12. 12.
    Anisov S. S., “Flip equivalence of surface triangulations,” Mosc. Univ. Math. Bull., 49, No. 2, 55–60 (1994).MATHMathSciNetGoogle Scholar
  13. 13.
    Johannson K., Homotopy Equivalences of 3-Manifolds with Boundaries, Springer-Verlag, Berlin (1979) (Lecture Notes in Mathematics; V. 761).MATHGoogle Scholar
  14. 14.
    Ovchinnikov M. A., “Construction of simple spines of Waldhausen manifolds,” in: Proceedings of the International Conference “Low-Dimensional Topology and Combinatorial Group Theory” (Chelyabinsk, 1999) [in Russian], Inst. Math. of NAN Ukraine, Kiev, 2000, pp. 65–86.Google Scholar
  15. 15.
    Martelli B. and Petronio C., “Three-manifolds having complexity at most 9,” Experiment. Math., 10, No. 2, 207–236 (2001).MATHMathSciNetGoogle Scholar
  16. 16.
    Thurston W. P., The Geometry and Topology of 3-Manifolds, Princeton Univ., Princeton (1978) (Mimeographed Notes).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.Chelyabinsk State UniversityChelyabinskRussia
  2. 2.Institute of Mathematics and MechanicsEkaterinburgRussia

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