Konrad-zuse-zentrum F ¨ Ur Informationstechnik Berlin Enumeration and Random Realization of Triangulated Surfaces Enumeration and Random Realization of Triangulated Surfaces

We discuss different approaches for the enumeration of triangulated surfaces. In particular, we enumerate all triangulated surfaces with 9 and 10 vertices. We also show how geometric realizations of orientable surfaces with few vertices can be obtained by choosing coordinates randomly.

[1]  S. Drake,et al.  Galileo's Discovery of the Parabolic Trajectory , 1975 .

[2]  Lukas Fins hiy,et al.  Generation of Oriented Matroids , 2002 .

[3]  Amos Altshuler.  Combinatorial 3-Manifolds with Few Vertices , 1974, J. Comb. Theory, Ser. A.

[4]  E. Steinitz,et al.  Vorlesungen über die Theorie der Polyeder unter Einfluss der Elemente der Topologie , 1934 .

[5]  S. Hougardy,et al.  Surface realization with the intersection edge functional , 2006, math/0608538.

[6]  P. J. Heawood.  Map-Colour Theorem , 1949 .

[7]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[8]  Peter Schuchert,et al.  Neighborly 2-Manifolds with 12 Vertices , 1996, J. Comb. Theory, Ser. A.

[9]  Jorge Nuno Silva,et al.  Mathematical Games , 1959, Nature.

[10]  F. Lutz,et al.  Triangulated Manifolds with Few Vertices: Vertex-Transitive Triangulations I , 2005, math/0506520.

[11]  W. T. Tutte,et al.  A Census of Planar Triangulations , 1962, Canadian Journal of Mathematics.

[12]  G. Ziegler.  Lectures on Polytopes , 1994 .

[13]  Ph. Furtwängler,et al.  Encyklopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen: Des Sechsten Bandes Erster Teil Geodäsie und Geophysik , 1906 .

[15]  A maximally symmetric polyhedron of genus 3 with 10 vertices , 1987 .

[16]  G. Ringel,et al.  Minimal triangulations on orientable surfaces , 1980 .

[17]  António Guedes de Oliveira,et al.  On the Generation of Oriented Matroids , 2000, Discret. Comput. Geom..

[18]  Thom Sulanke.  Irreducible triangulations of low genus surfaces , 2006 .

[19]  Jürgen Bokowski,et al.  A polyhedron of genus 4 with minimal number of vertices and maximal symmetry , 1989 .

[20]  D. Grace.  Computer search for non-isomorphic convex polyhedra. , 1965 .

[21]  B. Sturmfels.  Computational Synthetic Geometry , 1989 .

[22]  David W. Barnette,et al.  Generating the triangulations of the projective plane , 1982, J. Comb. Theory, Ser. B.

[23]  F. Enriques,et al.  Encyklopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen: Dritter Band: Geometrie , 1910 .

[24]  Frank H. Lutz,et al.  Isomorphism-free lexicographic enumeration of triangulated surfaces and 3-manifolds , 2006, Eur. J. Comb..

[25]  D. Barnette,et al.  All 2-manifolds have finitely many minimal triangulations , 1989 .

[26]  Amos Altshuler,et al.  An enumeration of combinatorial 3-manifolds with nine vertices , 1976, Discret. Math..

[27]  S. Lawrencenko.  Irreducible triangulations of a torus , 1987 .

[28]  Amos Altshuler,et al.  Neighborly 4-Polytopes with 9 Vertices , 1973, J. Comb. Theory, Ser. A.

[29]  Wolfgang Kühnel,et al.  Neighborly combinatorial 3-manifolds with dihedral automorphism group , 1985 .

[30]  Lars Schewe.  Satisfiability Problems in Discrete Geometry , 2007 .

[31]  G. C. Shephard,et al.  Convex Polytopes , 1969, The Mathematical Gazette.

[32]  G. Ringel,et al.  Wie man die geschlossenen nichtorientierbaren Flächen in möglichst wenig Dreiecke zerlegen kann , 1955 .

[33]  Jürgen Bokowski.  On Heuristic Methods for Finding Realizations of Surfaces , 2008 .

[34]  Thom Sulanke.  Generating irreducible triangulations of surfaces , 2006 .

[35]  Seiya Negami,et al.  Irreducible Triangulations of the Klein Bottle , 1997, J. Comb. Theory, Ser. B.

[36]  Thom Sulanke.  Note on the irreducible triangulations of the Klein bottle , 2006, J. Comb. Theory, Ser. B.

[37]  S. Fisk,et al.  Generations of triangulations of the sphere , 1967 .

[38]  Basudeb Datta,et al.  Two-dimensional weak pseudomanifolds on eight vertices , 2002 .

[40]  Dan Archdeacon,et al.  How to Exhibit Toroidal Maps in Space , 2007, Discret. Comput. Geom..