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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]
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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 

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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