Journal of the Mathematical Society of Japan

Classification of 3-bridge spheres of 3-bridge arborescent links

Yeonhee JANG

Full-text: Open access


In this paper, we give an isotopy classification of 3-bridge spheres of 3-bridge arborescent links, which are not Montesinos links. To this end, we prove a certain refinement of a theorem of J. S. Birman and H. M. Hilden [3] on the relation between bridge presentations of links and Heegaard splittings of 3-manifolds. In the proof of this result, we also give an answer to a question by K. Morimoto [23] on the classification of genus-2 Heegaard splittings of certain graph manifolds.

Article information

J. Math. Soc. Japan, Volume 65, Number 1 (2013), 97-136.

First available in Project Euclid: 24 January 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M12: Special coverings, e.g. branched

3-bridge spheres arborescent links


JANG, Yeonhee. Classification of 3-bridge spheres of 3-bridge arborescent links. J. Math. Soc. Japan 65 (2013), no. 1, 97--136. doi:10.2969/jmsj/06510097.

Export citation


  • J. S. Birman, Braids, Links, and Mapping Class Groups, Ann. of Math. Stud., 82, Princeton University Press, Princeton, N.J.
  • J. S. Birman, F. González-Acuña and J. M. Montesinos, Heegaard splittings of prime 3-manifolds are not unique, Michigan Math. J., 23 (1976), 97–103.
  • J. S. Birman and H. M. Hilden, Heegaard splittings of branched coverings of $S^3$, Trans. Amer. Math. Soc., 213 (1975), 315–352.
  • M. Boileau, D. J. Collins and H. Zieschang, Genus 2 Heegaard decompositions of small Seifert manifolds, Ann. Inst. Fourier (Grenoble), 41 (1991), 1005–1024.
  • M. Boileau and J.-P. Otal, Groupe des difféotopies de certaines variétés de Seifert, C. R. Acad. Sci. Paris Sér I Math., 303 (1986), 19–22. Groupes d'homéotopies et scindements de Heegaard des petites variétés de Seifert, Invent. Math., 106 (1991), 85–107.
  • M. Boileau and J. Porti, Geometrization of 3-orbifolds of cyclic type, Astérisque, 272 (2001), 208 pp.
  • M. Boileau and H. Zieschang, Nombre de ponts et générateurs méridiens des entrelacs de Montesinos, Comment. Math. Helv., 60 (1985), 270–279.
  • M. Boileau and B. Zimmermann, The $\pi$-orbifold group of a link, Math. Z., 200 (1989), 187–208.
  • M. Boileau and B. Zimmermann, Symmetries of nonelliptic Montesinos links, Math. Ann., 277 (1987), 563–584.
  • F. Bonahon and L. C. Siebenmann, New geometric splittings of classical knots and the classification and symmetries of arborescent knots, preprint.
  • J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, In: Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), Pergamon Press, Oxford, 1970, pp.,329–358.
  • D. Cooper, C. D. Hodgson and S. P. Kerckhoff, Three-Dimensional Orbifolds and Cone-Manifolds, MSJ Memoirs, 5, Math. Soc. Japan, Tokyo, 2000.
  • W. D. Dunbar, Geometric orbifolds, Rev. Mat. Univ. Complut. Madrid, 1 (1988), 67–99.
  • D. Futer and F. Guéritaud, Angled decompositions of arborescent link complements, Proc. Lond. Math. Soc. (3), 98 (2009), 325–364.
  • D. Gabai, Genera of the arborescent links, Mem. Amer. Math. Soc., 59, 1986.
  • Y. Jang, Three-bridge links with infinitely many three-bridge spheres, Topology Appl., 157 (2010), 165–172.
  • Y. Jang, Classification of 3-bridge arborescent links, Hiroshima Math. J., 41 (2011), 89–136.
  • K. Johannson, Homotopy Equivalences of 3-Manifolds with Boundaries, Lecture Notes in Math., 761, Springer-Verlag, Berlin, 1979.
  • T. Kobayashi, Structures of the Haken manifolds with Heegaard splittings of genus two, Osaka J. Math., 21 (1984), 437–455.
  • R. C. Lyndon and P. E. Schupp, Combinatorial group theory, Ergeb. Math. Grenzgeb, 89, Springer-Verlag, Berlin, Heidelberg, New York, 1977.
  • J. M. Montesinos, Classical Tessellations and Three-Manifolds, Universitext, Springer-Verlag, Berlin, 1987.
  • J. W. Morgan and H. Bass (Eds.), The Smith conjecture, Papers presented at the symposium held at Columbia University, New York, 1979, Pure Appl. Math., 112, Academic Press, 1984.
  • K. Morimoto, On minimum genus Heegaard splittings of some orientable closed 3-manifolds, Tokyo J. Math., 12 (1989), 321–355.
  • K. Morimoto and M. Sakuma, On unknotting tunnels for knots, Math. Ann., 289 (1991), 143–167.
  • K. Morimoto, M. Sakuma and Y. Yokota, Identifying tunnel number one knots, J. Math. Soc. Japan, 48 (1996), 667–688.
  • J.-P. Otal, Présentations en ponts du nœ ud trivial, C. R. Acad. Sci. Paris Sér. I Math., 294 (1982), 553–556.
  • J.-P. Otal, Présentations en ponts des nœ ud rationnels, In: Low-Dimensional Topology (Chelwood Gate, 1982), London Math. Soc. Lecture Note Ser., 95, Cambridge University Press, Cambridge, 1985, pp.,143–160.
  • M. Ozawa, Nonminimal bridge positions of torus knots are stabilized, Math. Proc. Cambridge Philos. Soc., 151 (2011), 307–317.
  • M. Sakuma, The geometries of spherical Montesinos links, Kobe J. Math., 7 (1990), 167–190.
  • M. Scharlemann and M. Tomova, Uniqueness of bridge surfaces for 2-bridge knots, Math. Proc. Cambridge Philos. Soc., 144 (2008), 639–650.
  • H. Schubert, Knoten mit zwei Brücken, Math. Z., 65 (1956), 133–170.
  • R. Weidmann, Some 3-manifolds with 2-generated fundamental group, Arch. Math. (Basel), 81 (2003), 589–595.