• Surfaces
  
  
• 3-Manifolds
  
  
• Vertex-Transitive Triangulations
  
  
• Further Examples
  
  
• Publications
  
  
• Software
  
  

Numbers of triangulated surfaces with  n  vertices

Numbers of triangulations of the orientable surfaces of genus  g  with  n  vertices

Numbers of triangulations of the non-orientable surfaces of genus  g  with  n  vertices

• Gunnar Brinkmann and Brendan D. McKay.   Fast generation of planar graphs.   MATCH Commun. Math. Comput. Chem. 58, 323-357 (2007).

• Thom Sulanke.   Counts of triangulations of surfaces.   2005.

• Frank H. Lutz.   Enumeration and random realization of triangulated surfaces.   Discrete Differential Geometry (A. I. Bobenko, P. Schröder, J. M. Sullivan, and G. M. Ziegler, eds.). Oberwolfach Seminars 38, 235-253. Birkhäuser, Basel, 2008.

• Thom Sulanke and Frank H. Lutz.   Isomorphism-free lexicographic enumeration of triangulated surfaces and 3-manifolds.   Eur. J. Comb. 30, 1965-1979 (2009).

• Thom Sulanke.   C program   lextri   (Version 0.15, 2007).

• Mark N. Ellingham and Chris Stephens.   Triangular embeddings of complete graphs (neighborly maps) with 12 and 13 vertices.   J. Combin. Des. 13, 336-344 (2005).

Equivelar surfaces

• Thom Sulanke and Frank H. Lutz.   Isomorphism-free lexicographic enumeration of triangulated surfaces and 3-manifolds.   Eur. J. Comb. 30, 1965-1979 (2009).

Polyhedral surfaces

• Frank H. Lutz.   Enumeration and random realization of triangulated surfaces.   Discrete Differential Geometry (A. I. Bobenko, P. Schröder, J. M. Sullivan, and G. M. Ziegler, eds.). Oberwolfach Seminars 38, 235-253. Birkhäuser, Basel, 2008.
• Stefan Hougardy, Frank H. Lutz, and Mariano Zelke.   Polyhedra of genus 2 with 10 vertices and minimal coordinates.   Electronic Geometry Models, No. 2005.08.001 (2007).
• Stefan Hougardy, Frank H. Lutz, and Mariano Zelke.   Polyhedra of genus 3 with 10 vertices and minimal coordinates.   Electronic Geometry Models, No. 2006.02.001 (2007).
• Stefan Hougardy, Frank H. Lutz, and Mariano Zelke.   Polyhedral tori with minimal integer coordinates.   Electronic Geometry Models, No. 2008.10.001 (2008).
• Stefan Hougardy, Frank H. Lutz, and Mariano Zelke.   Surface realization with the intersection edge functional.   Preprint, 22 pages, 2009; arXiv:math/0608538v2; Exp. Math., to appear.
• Frank H. Lutz and Nikolaus Witte.   Knotted polyhedral tori.   Preprint, 10 pages, 2007; arXiv:0707.1281.

Numbers of triangulated 3-manifolds with  n  vertices

• Frank H. Lutz.   Combinatorial 3-manifolds with 10 vertices.   Beitr. Algebra Geom. 49, 97-106 (2008).

• Thom Sulanke and Frank H. Lutz.   Isomorphism-free lexicographic enumeration of triangulated surfaces and 3-manifolds.   Eur. J. Comb. 30, 1965-1979 (2009).

• Thom Sulanke.   C program   lextet.

Small triangulations of geometric 3-manifolds

• S2xR_spaces  with  homology groups  and smallest known f_vectors
• spherical_3manifolds  with  homology groups  and smallest known f_vectors
• flat_3manifolds  with  homology groups  and smallest known f_vectors
• nil_3manifolds  with  homology groups  and smallest known f_vectors
• H2xR_spaces  with  homology groups  and smallest known f_vectors
• homology_3spheres  of geometry ~SL(2,r) with  homology groups  and smallest known f_vectors
• hyperbolic_dodecahedral_space  of Weber and Seifert with  homology groups  and smallest known f_vector
• hyperbolic_3manifolds  with  homology groups  and smallest known f_vectors

• connected sums  with  homology groups  and smallest known f_vectors

• Frank H. Lutz, Thom Sulanke, and Ed Swartz.   f-vectors of 3-manifolds.   Electron. J. Comb. 16, No. 2, Research Paper R13, 33 p. (2009).

• Frank H. Lutz.   Triangulated Manifolds with Few Vertices: Geometric 3-Manifolds.   Preprint, 48 pages, 2003; arXiv:math/0311116.

Simplicial manifolds with small valence

• Frank H. Lutz and John M. Sullivan.   Simplicial manifolds with small valence.   In preparation.

• Frank H. Lutz.   Triangulated surfaces and higher-dimensional manifolds.   Oberwolfach Reports 3, No. 1, 706-707 (2006).

• Frank H. Lutz.   Periodic foams and simplicial manifolds with small valence.   Oberwolfach Reports 4, No. 1, 228-230 (2007).

Vertex-transitive combinatorial manifolds with up to 15 vertices
(complete for d=2,3,9,10,11,12 and up to 13 vertices otherwise)

• 2-manifolds  with  homology groups  and  topological type
• 3-manifolds  with  homology groups  and  topological type
• 4-manifolds  with  homology groups  and  topological type
• 5-manifolds  with  homology groups  and  topological type
• 6-manifolds  with  homology groups  and  topological type
• 7-manifolds  with  homology groups  and  topological type
• 8-manifolds  with  homology groups  and  topological type
• 9-manifolds  with  homology groups  and  topological type
• 10-manifolds  with  homology groups  and  topological type
• 11-manifolds  with  homology groups  and  topological type

Vertex-transitive combinatorial pseudomanifolds with up to 15 vertices
(complete for d=3 and up to 13 vertices for d=4,5,6,7)

• 3-pseudomanifolds (.gz)  with  homology groups
• 5-pseudomanifolds  with  homology groups  and  topological type of the link

Nearly neighborly centrally symmetric (nncs) combinatorial spheres with vertex-transitive cyclic symmetry
(up to 22 vertices for d=3, up to 16 vertices for d=5, and up to 18 vertices for d=7)

• nncs-3-spheres
• nncs-5-spheres
• nncs-7-spheres

Centrally symmetric d-dimensional combinatorial products of spheres with vertex-transitive
dihedral symmetry (on n=2d+4 vertices for d=2,...,8) or cyclic symmetry (on n=2d+4 vertices for d=2,...,7)

• products_of_spheres  with  homology groups  and  topological type

Small triangulations of some well-known 4-manifolds

• 4-manifolds  with  homology groups  and smallest known f_vectors

Vertex-minimal triangulations

• RP^3
• L(3,1)
• CP^2
• RP^4
• K3 surface
• S^3 x S^2  with  Altshuler-Steinberg determinants

``Non''-spheres

• Barnette's_non polytopal 3-sphere
• Gruenbaum-Sreedharan's non-polytopal 3-sphere
• non-shellable, non-constructible 3-sphere
• Poincaré 3-sphere
• non-PL 5-sphere

• Bruno Benedetti and Frank H. Lutz.   The dunce hat and a minimal non-extendably collapsible 3-ball.   Preprint, 7 pages, 2009; arXiv:0912.3723.
• Anders Björner and Frank H. Lutz.   Simplicial manifolds, bistellar flips and a 16-vertex triangulation of the Poincaré homology 3-sphere.   Exp. Math. 9, 275-289 (2000).
• Anders Björner and Frank H. Lutz.   A 16-vertex triangulation of the Poincaré homology 3-sphere and non-PL spheres with few vertices.   Electronic Geometry Models, No. 2003.04.001 (2003).
• Péter Csorba and Frank H. Lutz.   Graph coloring manifolds.   Algebraic and Geometric Combinatorics, Euroconf. Math., Anogia, Crete, Greece, 2005 (C. A. Athanasiadis, V. V. Batyrev, D. I. Dais, M. Henk, and F. Santos, eds.). Contemporary Mathematics 423, 51-69. American Mathematical Society, Providence, RI, 2006.
• Michael Joswig and Frank H. Lutz.   One-Point Suspensions and Wreath Products of Polytopes and Spheres.   J. Comb. Theory, Ser. A 110, 193-216 (2005).
• Wolfgang Kühnel and Frank H. Lutz.   A census of tight triangulations.   Periodica Math. Hung. 39, 161-183 (1999).
• Frank H. Lutz.   Császár's Torus.   Electronic Geometry Models, No. 2001.02.069 (2002).
• Frank H. Lutz.   A vertex-minimal non-shellable simplicial 3-ball with 9 vertices and 18 facets.   Electronic Geometry Models, No. 2003.05.004 (2004).
• Frank H. Lutz.   Vertex-minimal not vertex-decomposable balls.   Electronic Geometry Models, No. 2003.06.001 (2004).
• Frank H. Lutz.   Small examples of non-constructible simplicial balls and spheres.   SIAM J. Discrete Math. 18, 103-109 (2004).
• Frank H. Lutz.   Triangulated Manifolds with Few Vertices and Vertex-Transitive Group Actions.   Dissertation. Shaker Verlag, Aachen, 1999.
• Frank H. Lutz.   Triangulated Manifolds with Few Vertices: Geometric 3-Manifolds.   Preprint, 48 pages, 2003; arXiv:math/0311116.
• Frank H. Lutz.   Triangulated Manifolds with Few Vertices: Centrally Symmetric Spheres and Products of Spheres.   Preprint, 26 pages, 2004; arXiv:math/0404465.
• Frank H. Lutz.   Triangulated Manifolds with Few Vertices: Combinatorial Manifolds.   Preprint, 37 pages, 2005; arXiv:math/0506372.
• Ekkehard G. Köhler and Frank H. Lutz.   Triangulated Manifolds with Few Vertices: Vertex-Transitive Triangulations I.   Preprint, 74 pages, 2005; arXiv:math/0506520.

Topological Software

• homology-3.0:  (pascal/C/C++)-progam by Frank Heckenbach;
• homology:  a (proposed) GAP share package by Jean-Guillaume Dumas, Frank Heckenbach, B. David Saunders, and Volkmar Welker;
• application TOPAZ (TOPology Application Zoo) of the polymake system:  software package by Ewgenij Gawrilow and Michael Joswig et al.

• FundamentalGroup (Version Nov/2003):  GAP routine by Frank H. Lutz to compute a ``small´´ presentation of the fundamental group of a finite simplicial complex.

• application TOPAZ (TOPology Application Zoo) of the polymake system:  software package by Ewgenij Gawrilow and Michael Joswig et al.

• lextri   (Version 0.15, 2007)   by  Thom Sulanke;
• lextet  (Version 0.24, 2007)   by  Thom Sulanke

• BISTELLAR (Version Nov/2003):  heuristics to reduce the size of a given triangulated manifold; it can be used to recognize combinatorial d-spheres and therefore to test whether a pure simplicial complex is a combinatorial manifold;
• BISTELLAR_EQUIVALENT (Version Feb/1999);
• MANIFOLD_VT (Version Apr/2002);
• SEIFERT (Version Nov/2003):  produces triangulations of all Seifert manifolds.

• JavaView:  by Konrad Polthier et al.

Last modified: Mon Jan 4 16:59:04 CET 2010