×

A census of cusped hyperbolic 3-manifolds. (English) Zbl 0910.57006

Each noncompact (or cusped) hyperbolic 3-manifold of finite volume can be decomposed into a finite collection of ideal hyperbolic tetrahedra. In the present work, a census is given for all hyperbolic 3-manifolds which can be obtained by gluing the faces of at most seven ideal tetrahedra. There are 6075 such manifolds, 4815 of them are orientable. The 103 manifolds obained from four or fewer tetrahedra are listed in an appendix to the paper, the others are in tables included on a microfiche supplement. In analogy with the enumeration of knots and links, each manifold is given a name indicating the number of ideal tetrahedra, the number of orientable and non-orientable cusps and finally its position in terms of increasing volume. For each manifold the following data are listed: volume, Chern-Simons invariant (if orientable), homology, symmetry or isometry group, shortest geodesic, chirality and a string of letters (code) from which the gluing pattern can be reconstructed. A description is given of how the enumeration has been carried out. As combinatorially there are too many gluing patterns already for a small number of tetrahedra, in a first step one has to find effective ways of eliminating a large number of gluings which could not possibly yield hyperbolic manifolds. Then in a second step computer programs as SnapPea are used to determine which of the remaining gluings in fact admit hyperbolic structures, to remove duplicates from the lists and to compute the various invariants. It is a hope that the lists may give hints on the distribution and on possible classification schemes for hyperbolic 3-manifolds, besides giving examples of small hyperbolic 3-manifolds with certain specific properties.

MSC:

57M50 General geometric structures on low-dimensional manifolds
57-04 Software, source code, etc. for problems pertaining to manifolds and cell complexes
57N10 Topology of general \(3\)-manifolds (MSC2010)

Software:

SnapPea
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Colin C. Adams, The noncompact hyperbolic 3-manifold of minimal volume, Proc. Amer. Math. Soc. 100 (1987), no. 4, 601 – 606. · Zbl 0634.57008
[2] Colin Adams and William Sherman, Minimum ideal triangulations of hyperbolic 3-manifolds, Discrete Comput. Geom. 6 (1991), no. 2, 135 – 153. · Zbl 0722.52008
[3] J. Emert, private communication.
[4] D. B. A. Epstein and R. C. Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Differential Geom. 27 (1988), no. 1, 67 – 80. · Zbl 0611.53036
[5] Martin Hildebrand and Jeffrey Weeks, A computer generated census of cusped hyperbolic 3-manifolds, Computers and mathematics (Cambridge, MA, 1989) Springer, New York, 1989, pp. 53 – 59. · Zbl 0674.57001
[6] Sóstenes Lins, Gems, computers and attractors for 3-manifolds, Series on Knots and Everything, vol. 5, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. · Zbl 0868.57002
[7] S. Matveev and A. Fomenko, Constant energy surfaces of Hamiltonian systems, enumeration of three-dimensional manifolds in increasing order of complexity, and computation of volumes of closed hyperbolic 3-manifolds, Russian Math. Surveys 43 (1988), 3-24. · Zbl 0671.58008
[8] Robert Meyerhoff, Density of the Chern-Simons invariant for hyperbolic 3-manifolds, Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 112, Cambridge Univ. Press, Cambridge, 1986, pp. 217 – 239. Robert Meyerhoff, Hyperbolic 3-manifolds with equal volumes but different Chern-Simons invariants, Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 112, Cambridge Univ. Press, Cambridge, 1986, pp. 209 – 215.
[9] Robert Myers, Simple knots in compact, orientable 3-manifolds, Trans. Amer. Math. Soc. 273 (1982), no. 1, 75 – 91. · Zbl 0508.57008
[10] Peter Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401 – 487. · Zbl 0561.57001
[11] W. Thurston, The geometry and topology of 3-manifolds, Princeton Lecture Notes. · Zbl 0483.57007
[12] William P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy. · Zbl 0873.57001
[13] J. Weeks, SnapPea: A computer program for creating and studying hyperbolic 3-manifolds, available by anonymous ftp from geom.umn.edu/pub/software/snappea/.
[14] Jeffrey R. Weeks, Convex hulls and isometries of cusped hyperbolic 3-manifolds, Topology Appl. 52 (1993), no. 2, 127 – 149. · Zbl 0808.57005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.