SnapPea
A program for creating and studying hyperbolic 3-manifolds. 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.
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References in zbMATH (referenced in 152 articles )
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Sorted by year (- Vogeler, Roger: Cusp transitivity in hyperbolic 3-manifolds (2021)
- Kolpakov, Alexander; Reid, Alan W.; Riolo, Stefano: Many cusped hyperbolic 3-manifolds do not bound geometrically (2020)
- Paoluzzi, Luisa; Sakuma, Makoto: Prime amphicheiral knots with free period 2 (2020)
- Sbrodova, E. A.; Tarkaev, V. V.; Fominykh, E. A.; Shumakova, E. V.: Virtual 3-manifolds of complexity 1 and 2 (2019)
- Trnková, Maria: Rigorous computations with an approximate Dirichlet domain (2019)
- Cristofori, Paola; Fominykh, Evgeny; Mulazzani, Michele; Tarkaev, Vladimir: Minimal 4-colored graphs representing an infinite family of hyperbolic 3-manifolds (2018)
- Vesnin, A. Yu.; Matveev, S. V.; Fominykh, E. A.: New aspects of complexity theory for 3-manifolds (2018)
- Cristofori, Paola; Fominykh, Evgeny; Mulazzani, Michele; Tarkaev, Vladimir: 4-colored graphs and knot/link complements (2017)
- Matignon, Daniel: Hyperbolic (H)-knots in non-trivial lens spaces are not determined by their complement (2017)
- Toth, Csaba D. (ed.); Goodman, Jacob E. (ed.); O’Rourke, Joseph (ed.): Handbook of discrete and computational geometry (2017)
- Garoufalidis, Stavros: The 3D index of an ideal triangulation and angle structures (2016)
- Hodgson, Craig D.; Issa, Ahmad; Segerman, Henry: Non-geometric veering triangulations (2016)
- Kawagoe, Kenichi: On the formulae for the colored HOMFLY polynomials (2016)
- Conner, Gregory R.; Meilstrup, Mark; Repovš, Dušan: The geometry and fundamental groups of solenoid complements (2015)
- Garoufalidis, Stavros; Goerner, Matthias; Zickert, Christian: Gluing equations for (\mathrmPGL(n, \mathbbC))-representations of 3-manifolds (2015)
- Gendulphe, Matthieu: Systole and inradius of noncompact hyperbolic manifolds (2015)
- Millichap, Christian: Factorial growth rates for the number of hyperbolic 3-manifolds of a given volume (2015)
- Vesnin, A. Yu.; Tarkaev, V. V.; Fominykh, E. A.: Cusped hyperbolic 3-manifolds of complexity 10 having maximum volume (2015)
- Baker, Kenneth L.: Counting genus one fibered knots in Lens spaces (2014)
- Baker, Kenneth L.; Guntel Doleshal, Brandy; Hoffman, Neil: On manifolds with multiple lens space fillings (2014)