Glasnik Matematicki, Vol. 48, No. 1 (2013), 173-183.

ON IDENTIFYING HYPERBOLIC 3-MANIFOLDS AS LINK COMPLEMENTS IN THE 3-SPHERE

Dubravko Ivanšić

Abstract.   We give a straightforward method that helps recognize when a noncompact hyperbolic 3-manifold is a link complement in the 3-sphere and automatically produces the link diagram. The method is based on converting a side-pairing to a handle decomposition.

2010 Mathematics Subject Classification.   57M50, 57M25.

Key words and phrases.   Handle decomposition, hyperbolic manifold, link complement.

DOI: 10.3336/gm.48.1.14

References:

1. S. Bleiler and C. Hodgson, Spherical space forms and Dehn filling, Topology 35 (1996), 809-833.
   MathSciNet     CrossRef

2. P. J. Callahan, J. C. Dean and J. R. Weeks, The simplest hyperbolic knots, J. Knot Theory Ramifications 8 (1999), 279-297.
   MathSciNet     CrossRef

3. P. J. Callahan and A. W. Reid, Hyperbolic structures on knot complements, Chaos Solitons Fractals 9 (1998), 705-738.
   MathSciNet     CrossRef

4. G. K. Francis, A topological picturebook, Springer-Verlag, 1987.
   MathSciNet    

5. R. Gompf and A. Stipsicz, 4-manifolds and Kirby calculus, AMS, Providence, 1999.
   MathSciNet    

6. D. Ivanšić, J. Ratcliffe and S. Tschantz, Complements of tori and Klein bottles in the 4-sphere that have hyperbolic structure, Algebr. Geom. Topol. 5 (2005), 999-1026.
   MathSciNet     CrossRef

7. D. Ivanšić, A topological 4-sphere that is standard, to appear in Adv. Geom.

8. D. Ivanšić, Hyperbolic structure on a complement of tori in the 4-sphere, Adv. Geom. 4 (2004), 119-139.
   MathSciNet     CrossRef

9. R. Riley, Discrete parabolic representations of link groups, Mathematika 22 (1975), 141-150.
   MathSciNet     CrossRef

10. R. Riley, A quadratic parabolic group, Math. Proc. Cambridge Philos. Soc. 77 (1975), 281-288.
    MathSciNet     CrossRef

11. R. Riley, Seven excellent knots, in: Low-dimensional topology (Bangor, 1979), Cambridge Univ. Press, Cambridge, 1982, 81-151.
    MathSciNet     CrossRef

12. J. Ratcliffe and S. Tschantz, The volume spectrum of hyperbolic 4-manifolds, Experiment. Math. 9 (2000), 101-125.
    MathSciNet     CrossRef

13. W. Thurston, The geometry and topology of three-manifolds, Princeton University lecture notes, 1979, 1982.

14. W. P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton University Press, Princeton, 1997, edited by Silvio Levy.
    MathSciNet    

15. J. Weeks, Snappea: a computer program for creating and studying hyperbolic 3-manifolds, available at http://www.geometrygames.org/SnapPea/.

16. N. Wielenberg, The structure of certain subgroups of the Picard group, Math. Proc. Cambridge Philos. Soc. 84 (1978), 427-436.
    MathSciNet     CrossRef