Skip to main content
SpringerLink
Log in

Around the Borromean link

En torno al enlace de Borromeo

  • Published:
RACSAM - Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas Aims and scope Submit manuscript

Abstract

This is a survey of some consequences of the fact that the fundamental group of the orbifold with singular set the Borromean link and isotropy cyclic of order 4 is a universal kleinian group.

Resumen

Se presenta una panorámica de lo que se ha podido deducir hasta ahora del hecho de ser universal el grupo fundamental de los anillos de Borromeo con isotropía 4.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Armstrong, M. A., (1968). The fundamental group of the orbit space of a discontinuous group.Proc. Cambridge Philos. Soc.,64, 299–301.

    Article  MATH  MathSciNet  Google Scholar 

  2. Cromwell, Peter, Beltrami, Elisabetta and Rampichini, Marta, (1998). The Borromean rings,Math. Intelligencer,20, 1, 53–62.

    Article  MATH  MathSciNet  Google Scholar 

  3. Fox, R.-H., (1962). A quick trip through knot theory.Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), 120–167, Prentice-Hall, Englewood Cliffs, N.J.55.20

    Google Scholar 

  4. Hilden, Hugh M., Lozano, M. T. and Montesinos, José María, (1983). Universal knots.Bull. Amer. Math. Soc., (N.S.),8, 3, 449–450.

    Article  MATH  MathSciNet  Google Scholar 

  5. Hilden, Hugh M., Lozano, María Teresa and Montesinos, JoséMaría, Universal knots.Knot theory and manifolds (Vancouver, B.C., 1983), 25–59, Lecture Notes in Math., 1144, Springer, Berlin, 1985.

    Google Scholar 

  6. Hilden, HughM., Lozano, María Teresa and Montesinos, JoséMaría, (1983). TheWhitehead link, the Borromean rings and the knot 946 are universal.Collect. Math.,34, 1, 19–28.

    MATH  MathSciNet  Google Scholar 

  7. Hilden, Hugh M., Lozano, María Teresa and Montesinos, José María, (1985). On knots that are universal.Topology,24, 4, 499–504.

    Article  MATH  MathSciNet  Google Scholar 

  8. Hilden, H. M., Lozano, M. T., Montesinos, J. M. and Whitten, W. C., (1987). On universal groups and threemanifolds,Invent. Math.,87, 3, 441–456.

    Article  MATH  MathSciNet  Google Scholar 

  9. Hilden, Hugh M., Lozano, María Teresa and Montesinos, José María, (1988). On the universal group of the Borromean rings,Differential topology (Siegen, 1987), 1–13, Lecture Notes in Math., 1350, Springer, Berlin, 1988.

    Google Scholar 

  10. Hilden, Hugh M., Lozano, María Teresa and Montesinos-Amilibia, José María, On the Borromean orbifolds: geometry and arithmetic. Topology ’90 (Columbus, OH, 1990), 133–167, Ohio State Univ. Math. Res. Inst. Publ., 1, de Gruyter, Berlin, 1992. (Reviewer: B. N. Apanasov)

    Google Scholar 

  11. Hilden, Hugh M., Lozano, María Teresa and Montesinos-Amilibia, José María, (1993). Universal 2-bridge knot and link orbifolds.J. Knot Theory Ramifications,2, 2, 141–148.

    Article  MATH  MathSciNet  Google Scholar 

  12. Hilden, Hugh M., Lozano, María Teresa and Montesinos-Amilibia, José María, (2001). The arithmetic structure of a universal group. Dedicated to the memory of Professor M. Pezzana (Italian).Atti Sem. Mat. Fis. Univ. Modena,49, suppl., 1–14.

    MATH  MathSciNet  Google Scholar 

  13. Hilden, Hugh M., Lozano, María Teresa and Montesinos-Amilibia, José María, (2004). On 2-universal knots.Bol. Soc. Mat. Mexicana (3),10, Special Issue, 239–253.

    MATH  MathSciNet  Google Scholar 

  14. Hilden, Hugh M., Lozano, María Teresa and Montesinos-Amilibia, José María, (2006). On hyperbolic 3-manifolds with an infinite number of fibrations over S1,Math. Proc. Cambridge Philos. Soc.,140, 1, 79–93.

    Article  MATH  MathSciNet  Google Scholar 

  15. Jones, Kerry N., (1994). The structure of closed non-positively curved Euclidean cone 3-manifolds.Pacific J. Math.,163, 2, 297–313.

    MATH  MathSciNet  Google Scholar 

  16. Jones, Kerry N., (1994). Geometric structures on branched covers over universal links.Geometric topology (Haifa, 1992), 47–58, Contemp. Math., 164, Amer. Math. Soc., Providence, RI.

    Google Scholar 

  17. Lozano, María Teresa and Montesinos-Amilibia, José María, (1997). Geodesic flows on hyperbolic orbifolds, and universal orbifolds.Pacific J. Math.,177, 1, 109–147.

    Article  MATH  MathSciNet  Google Scholar 

  18. Maclachlan, Colin and Reid, Alan W.The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics,219. Springer-Verlag, New York, 2003. xiv+463

    MATH  Google Scholar 

  19. Matsumoto, K., (2006). Automorphic functions with respect to the fundamental group of the complement of the Borromean rings,J. Math. Sci. Univ. Tokyo,13, 1, 1–11.

    MATH  MathSciNet  Google Scholar 

  20. Matsumoto, Yukio and Montesinos-Amilibia, José María, (1991). A proof of Thurston’s uniformization theorem of geometric orbifolds.Tokyo J. Math.,14, 1, 181–196.

    Article  MATH  MathSciNet  Google Scholar 

  21. Montesinos, José María, (1983). Representing 3-manifolds by a universal branching set.Math. Proc. Cambridge Philos. Soc.,94, 1, 109–123.

    Article  MATH  MathSciNet  Google Scholar 

  22. Ramírez, Arturo, (1975). On a theorem of Alexander, (Spanish),An. Inst. Mat. Univ. Nac. Autónoma México,15, 1, 77–81.

    Google Scholar 

  23. Ratcliffe, John G., (2006).Foundations of hyperbolic manifolds. Second edition. Graduate Texts in Mathematics,149, Springer, New York, xii+779.

    MATH  Google Scholar 

  24. Rolfsen, Dale, (1990).Knots and links, Corrected reprint of the 1976 original. Mathematics Lecture Series, 7. Publish or Perish, Inc., Houston, TX, xiv+439 pp.

    MATH  Google Scholar 

  25. Toda, Masahito, (2004). Representation of finite groups and the first Betti number of branched coverings of a universal Borromean orbifold.Cent. Eur. J. Math.,2, 2, 218–249 (electronic).

    Article  MATH  MathSciNet  Google Scholar 

  26. Thurston, William P.,Three-dimensional geometry and topology, Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series,35. Princeton University Press, Princeton, NJ, 1997. x+311 pp.

    Google Scholar 

  27. Uchida, Yoshiaki, (1992). Universal pretzel links.Knots 90, (Osaka, 1990), 241–270, de Gruyter, Berlin.

    Google Scholar 

  28. Uchida, Yoshiaki, (1991). Universal chains,Kobe J. Math.,8, 1, 55–65.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Real Academia de Ciencias de Madrid, Madrid, Spain

    José María Montesinos-Amilibia

Authors
  1. José María Montesinos-Amilibia
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Montesinos-Amilibia, J.M. Around the Borromean link. Rev. R. Acad. Cien. Serie A. Mat. 102, 75–87 (2008). https://doi.org/10.1007/BF03191812

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03191812

Keywords

Mathematics Subject Classifications

Search

Navigation