Hiroshima Mathematical Journal

Classification of 3-bridge arborescent links

Yeonhee Jang

Full-text: Open access

Abstract

In this paper, we give a complete classification of 3-bridge arborescent links.

Article information

Source
Hiroshima Math. J., Volume 41, Number 1 (2011), 89-136.

Dates
First available in Project Euclid: 31 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.hmj/1301586291

Digital Object Identifier
doi:10.32917/hmj/1301586291

Mathematical Reviews number (MathSciNet)
MR2809049

Zentralblatt MATH identifier
1231.57006

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57N10: Topology of general 3-manifolds [See also 57Mxx]

Keywords
3-bridge presentations arborescent links algebraic links Heegaard splittings

Citation

Jang, Yeonhee. Classification of 3-bridge arborescent links. Hiroshima Math. J. 41 (2011), no. 1, 89--136. doi:10.32917/hmj/1301586291. https://projecteuclid.org/euclid.hmj/1301586291


Export citation

References

  • J. S. Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82. Princeton University Press, Princeton, N.J
  • J. S. Birman and H. Hilden, Heegaard splittings and 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 3-manifolds, Ann. Inst. Fourier (Grenoble) 41 (1991), 1005–1024.
  • M. Boileau and R. Weidmann, On invertible generating pairs of fundamental groups of graph manifolds, Geom. Topol. Monogr. 14 (2008), 105–128.
  • M. Boileau and H. Zieschang, Nombre de ponts et générateurs méridiens des entrelacs de Montesinos, Comment. Math. Helvetici 60 (1985), 270–279.
  • F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, available at http://www-bcf.usc.edu/~fbonahon.
  • G. Burde and H. Zieschang, Knots, Stud. Math. 5, Walter de Gruyter, 1985.
  • J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), Pergamon Press, Oxford, 1970, 329–358.
  • D. Futer and F. Guéritaud, Angled decompositions of arborescent link complements, Proc. Lond. Math. Soc. (3) 98 (2009), no. 2, 325–364.
  • D. Gabai, Genera of the arborescent links, Memoirs of the American Mathematical Society, 339, 1986.
  • C. Gordon and J. Luecke, Knots with unknotting number $1$ and essential Conway spheres, Algebr. Geom. Topol. 6 (2006), 2051–2116.
  • C. Hodgson and J.H. Rubinstein, Involutions and isotopies of lens spaces, Knot theory and manifolds (Vancouver, B.C., 1983), 60–96, Lecture Notes in Math., 1144 Springer, Berlin, 1985.
  • W. Jaco, Lectures on three manifold topology, CBMS Regional Conf. Ser. in Math., 43.
  • Y. Jang, Three-bridge links with infinitely many three-bridge spheres, Topology Appl. 157 (2010), 165–172.
  • K. Johannson, Homotopy equivalences of $3$-manifolds with boundaries, Lecture Notes in Mathematics, 761. Springer, Berlin, 1979. ii+303 pp.
  • A. Kawauchi, A survey of knot theory, Birkhäuser Verlag, Basel, 1996.
  • T. Kobayashi, Structures of the Haken manifolds with Heegaard splittings of genus two, Osaka J. Math. 21 (1984), 437–455.
  • J. M. Montesinos, Classical tessellations and three-manifolds, Universitext. Springer-Verlag, Berlin, 1987. xviii+230 pp.
  • K. Morimoto, On minimum genus Heegaard splittings of some orientable closed 3-manifolds, Tokyo J. Math. 12 (1989), 321–355.
  • U. Oertel, Closed incompressible surfaces in complements of star links, Pacific J. Math. 111 (1984), no. 1, 307–332.
  • J.-P. Otal, Présentations en ponts du nœ ud trivial, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 16, 553–556.
  • J.-P. Otal, Présentations en ponts des nœ ud rationnels, Low-dimensional topology (Chelwood Gate, 1982), 143–160, London Math. Soc. Lecture Note Ser., 95, Cambridge Univ. Press, Cambridge, 1985.
  • P. Orlik, Seifert manifolds, Lecture notes in Mathematics, 291 Springer-verlag, Berlin-New York, 1972.
  • M. Sakuma, Realization of the symmetry groups of links, Transformation groups (Osaka, 1987), 291–306, Lecture Notes in Math., 1375, Springer, Berlin, 1989.
  • M. Sakuma, The geometries of spherical Montesinos links, Kobe J. Math. 7 (1990), no. 2, 167–190.
  • M. Scharlemann and M. Tomova, Uniqueness of bridge surfaces for 2-bridge knots, Math. Proc. Cambridge Philos. Soc. 144 (2008), no. 3, 639–650.
  • H. Schubert, Knoten mit zwei Brücken, Math. Z. 7 (1956), 133–170.
  • M. Tollefson, Involutions of sufficiently large $3$-manifolds, Topology 20 (1981), no. 4, 323–352.
  • F. Waldhausen, Eine klasse von 3-dimensionalen mannigfaltigkeiten I, II, Invent. Math. 3 (1967), 308–333; ibid. 4 (1967), 87–117.
  • R. Weidmann, Some 3-manifolds with 2-generated fundamental group, Arch. Math. (Basel) 81 (2003), 589–595.
  • H. Zieschang, Classification of Montesinos knots, Topology Proc. Leningrad 1982, Lecture Notes in Math. 1060 Springer-Verlag (1984), 378–389.