Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-09T13:36:09.419Z Has data issue: false hasContentIssue false
  • English
  • Français

Automorphisms of the 3-sphere that preserve spatial graphs and handlebody-knots

Published online by Cambridge University Press:  03 February 2015

YUYA KODA
YUYA KODA*
Affiliation:
Department of Mathematics, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526, Japan. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the group of isotopy classes of automorphisms of the 3-sphere that preserve a spatial graph or a handlebody-knot embedded in it. We prove that the group is finitely presented for an arbitrary spatial graph or a reducible handlebody-knot of genus two. We also prove that the groups for “most” irreducible genus two handlebody-knots are finite.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 1 , July 2015 , pp. 1 - 22
Copyright
Copyright © Cambridge Philosophical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Akbas, E. A presentation for the automorphisms of the 3-sphere that preserve a genus two Heegaard splitting. Pacific J. Math. 236 (2008), no. 2, 201–222.Google Scholar
[2] Benedetti, R. and Frigerio, R. Levels of knotting of spatial handlebodies. Trans. Amer. Math. Soc. 365 (2013), no. 4, 2099–2167.Google Scholar
[3] Birman, J. S. Braids, links and mapping class groups. Ann. Math. Stud. 82 (Princeton University Press, 1974).Google Scholar
[4] Boileau, M. and Zimmermann, B. The π-orbifold group of a link. Math. Z. 200 (1989), no. 2, 187–208.Google Scholar
[5] Brown, K. S. Presentations for groups acting on simply-connected complexes. J. Pure Appl. Algebra 32 (1984), no. 1, 1–10.Google Scholar
[6] Cho, S. Homeomorphisms of the 3-sphere that preserve a Heegaard splitting of genus two. Proc. Amer. Math. Soc. 136 (2008), no. 3, 1113–1123.Google Scholar
[7] Cho, S. Genus-two Goeritz groups of lens spaces. Pacific J. Math. 265 (2013), no. 1, 1–16.Google Scholar
[8] Cho, S. and McCullough, D. The tree of knot tunnels. Geom. Topol. 13 (2009), no. 2, 769–815.Google Scholar
[9] Cooper, D., Hodgson, C. D. and Kerckhoff, S. P. Three-dimensional orbifolds and cone-manifolds. MSJ Memoirs 5 (Mathematical Society of Japan, 2000).Google Scholar
[10] Flapan, E., Naimi, R., Pommersheim, J. and Tamvakis, H. Topological symmetry groups of graphs embedded in the 3-sphere. Comment. Math. Helv. 80 (2005), no. 2, 317–354.Google Scholar
[11] Fomenko, A. T. and Matveev, S. V. Algorithmic and computer methods for three-manifolds. Math. Appl. 425 (Kluwer Academic Publishers, 1997).Google Scholar
[12] Goeritz, L. Die Abbildungen der Brezelfläche und der Vollbrezel vom Geschlecht 2. Abh. Math. Sem. Univ. Hamburg 9 (1933), no. 1, 244–259.Google Scholar
[13] Gordon, C. McA. On primitive sets of loops in the boundary of a handlebody. Topology Appl. 27 (1987), no. 3, 285–299.Google Scholar
[14] Gordon, C. McA. and Luecke, J. Knots are determined by their complements. J. Amer. Math. Soc. 2 (1989), no. 2, 371–415.Google Scholar
[15] Hamenstädt, U. and Hensel, S. The geometry of the handlebody groups I: distortion. J. Topol. Anal. 4 (2012), no. 1, 71–97.Google Scholar
[16] Heard, D., Hodgson, C., Martelli, B. and Petronio, C. Hyperbolic graphs of small complexity. Experiment. Math. 19 (2010), no. 2, 211–236.Google Scholar
[17] Holzmann, W. H. An equivariant torus theorem for involutions. Trans. Amer. Math. Soc. 326 (1991), no. 2, 887–906.Google Scholar
[18] Ishii, A. Moves and invariants for knotted handlebodies. Algebr. Geom. Topol. 8 (2008), no. 3, 1403–1418.Google Scholar
[19] Ishii, A., Kishimoto, K., Moriuchi, H. and Suzuki, M. A table of genus two handlebody-knots up to six crossings. J. Knot Theory Ramifications 21 (2012), no. 4, 1250035, 9 pp.Google Scholar
[20] Johannson, K. Homotopy equivalences of 3-manifolds with boundaries. Lecture Notes in Math. 761 (Springer-Verlag, 1979).Google Scholar
[21] Kapovich, M. Hyperbolic manifolds and discrete groups. Prog. Math. 183 (Birkhäuser Verlag, 2001).Google Scholar
[22] Kawauchi, A. A Survey of Knot Theory (Birkhäuser Verlag, 1996).Google Scholar
[23] Kojima, S. Isometry transformations of hyperbolic 3-manifolds. Topology Appl. 29 (1988), no. 3, 297–307.Google Scholar
[24] Lee, J. H. and Lee, S. Inequivalent handlebody-knots with homeomorphic complements. Algebr. Geom. Topol. 12 (2012), no. 2, 1059–1079.Google Scholar
[25] McCullough, D. Virtually geometrically finite mapping class groups of 3-manifolds. J. Differential Geom. 33 (1991), no. 1, 1–65.Google Scholar
[26] Motto, M. Inequivalent genus 2 handlebodies in S 3 with homeomorphic complement. Topology Appl. 36 (1990), no. 3, 283–290.Google Scholar
[27] Scharlemann, M. Heegaard Splittings of Compact 3-Manifolds. Handbook of geometric topology, 921–953 (North-Holland, 2002).Google Scholar
[28] Scharlemann, M. Automorphisms of the 3-sphere that preserve a genus two Heegaard splitting. Bol. Soc. Mat. Mexicana (3) 10 (2004), Special Issue, 503–514.Google Scholar
[29] Serre, J.-P. Arbres, amalgames, SL2. Astérisque 46 (Société Mathématique de France, 1977).Google Scholar
[30] Simon, J. Topological chirality of certain molecules. Topology 25 (1986), no. 2, 229–235.Google Scholar
[31] Sims, C. C. Computation with Finitely Presented Groups. Encyclopedia Math. Appl. 48, (Cambridge University Press, 1994).Google Scholar
[32] Waldhausen, F. Heegaard-Zerlegungen der 3-Sphäre. Topology 7 (1968), 195–203.Google Scholar