Isomorphism-free lexicographic enumeration of triangulated surfaces and 3-manifolds

We present a fast enumeration algorithm for combinatorial 2- and 3-manifolds. In particular, we enumerate all triangulated surfaces with 11 and 12 vertices and all triangulated 3-manifolds with 11 vertices. We further determine all equivelar polyhedral maps on the non-orientable surface of genus 4 as well as all equivelar triangulations of the orientable surface of genus 3 and the non-orientable surfaces of genus 5 and 6.

[1]  E. C. Zeeman,et al.  IMBEDDING OF MANIFOLDS IN EUCLIDEAN SPACE , 1961 .

[2]  Basudeb Datta,et al.  Degree-regular triangulations of torus and Klein bottle , 2004 .

[3]  Peter McMullen,et al.  Equivelar polyhedral manifolds inE3 , 1982 .

[4]  David Barnette.  All triangulations of the projective plane are geometrically realizable inE4 , 1983 .

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

[6]  Alexander Russell,et al.  Computational topology: ambient isotopic approximation of 2-manifolds , 2003, Theor. Comput. Sci..

[7]  Isabella Novik,et al.  A Note on Geometric Embeddings of Simplicial Complexes in a Euclidean Space , 2000, Discret. Comput. Geom..

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

[9]  Basudeb Datta,et al.  Equivelar Polyhedra with Few Vertices , 2001, Discret. Comput. Geom..

[10]  Gennaro Amendola,et al.  Decomposition and Enumeration of Triangulated Surfaces , 2007, Exp. Math..

[11]  Rainer Bodendiek,et al.  Topics in combinatorics and graph theory : essays in honour of Gerhard Ringel , 1990 .

[12]  Anita Patil-Deshmukh,et al.  Maps , 1992 .

[13]  G. Perelman.  Ricci flow with surgery on three-manifolds , 2003, math/0303109.

[14]  Ulrich Brehm,et al.  Polyhedral Maps with Few Edges , 1990 .

[15]  A. Upadhyay,et al.  Degree-regular triangulations of the double-torus , 2005, math/0508106.

[16]  Edwin E. Moise,et al.  Affine structures in 3-manifolds, V, The triangulation theorem and Hauptvermutung , 1952 .

[17]  Jörg M. Wills,et al.  On Coxeter's regular skew polyhedra , 1986, Discret. Math..

[18]  H. Whitney.  The Self-Intersections of a Smooth n-Manifold in 2n-Space , 1944 .

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

[20]  Amos Altshuler,et al.  Construction and enumeration of regular maps on the torus , 1973, Discret. Math..

[21]  J. Rubinstein,et al.  An Algorithm to Recognize the 3-Sphere , 1995 .

[22]  Lars Schewe,et al.  Nonrealizable Minimal Vertex Triangulations of Surfaces: Showing Nonrealizability Using Oriented Matroids and Satisfiability Solvers , 2008, Discret. Comput. Geom..

[23]  Aleksandar Mijatović.  Triangulations of Seifert fibred manifolds , 2003 .

[24]  Amos Altshuler.  Construction and Representation of Neighborly Manifolds , 1997, J. Comb. Theory, Ser. A.

[25]  Atsuhiro Nakamoto,et al.  Geometric Realization of a Triangulation on the Projective Plane with One Face Removed , 2008, Discret. Comput. Geom..

[26]  Jörg M. Wills,et al.  Geometric realizations for Dyck's regular map on a surface of genus 3 , 1986, Discret. Comput. Geom..

[27]  R. Read.  Every one a Winner or how to Avoid Isomorphism Search when Cataloguing Combinatorial Configurations , 1978 .

[28]  Felix Klein,et al.  Ueber die Transformation siebenter Ordnung der elliptischen Functionen , 1878 .

[29]  Peter McMullen,et al.  Polyhedral 2-manifolds inE3 with unusually large genus , 1983 .

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

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

[32]  B. Datta.  A Note on the Existence of {k, k}-equivelar Polyhedral Maps , 2005, math/0506618.

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

[34]  Wolfgang Kühnel,et al.  Equivelar maps on the torus , 2008, Eur. J. Comb..

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

[36]  J. Bokowski,et al.  All Realizations of Möbius' Torus with 7 Vertices , 1991 .

[37]  Joseph O'Rourke,et al.  Handbook of Discrete and Computational Geometry, Second Edition , 1997 .

[38]  B. Grünbaum,et al.  An enumeration of simplicial 4-polytopes with 8 vertices , 1967 .

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

[40]  A. Fomenko,et al.  THE PROBLEM OF DISCRIMINATING ALGORITHMICALLY THE STANDARD THREE-DIMENSIONAL SPHERE , 1974 .

[41]  Jonathan L. Gross,et al.  Handbook of graph theory , 2007, Discrete mathematics and its applications.

[42]  D. Timmreck.  Necessary Conditions for Geometric Realizability of Simplicial Complexes , 2007, 0705.1912.

[43]  A. Mijatović.  Simplifying triangulations of S3 , 2003 .

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

[45]  Brendan D. McKay,et al.  Isomorph-Free Exhaustive Generation , 1998, J. Algorithms.

[46]  Walther Dyck.  Ueber Aufstellung und Untersuchung von Gruppe und Irrationalität regulärer Riemann'scher Flächen , 1880 .

[47]  Peter McMullen,et al.  Infinite series of combinatorially regular polyhedra in three-space , 1988 .

[48]  Frank H. Lutz.  Small Examples of Nonconstructible Simplicial Balls and Spheres , 2004, SIAM J. Discret. Math..

[49]  Frank H. Lutz,et al.  Knotted Polyhedral Tori , 2007, 0707.1281.

[50]  Amos Altshuler.  Polyhedral realization in R3 of triangulations of the torus and 2-manifolds in cyclic 4-polytopes , 1971, Discret. Math..

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

[52]  W. T. Tutte.  How to Draw a Graph , 1963 .

[53]  Frank H. Lutz.  Combinatorial 3-manifolds with 10 vertices , 2006 .

[54]  Abigail Thompson,et al.  Thin Position and the Recognition Problem for $\bold{S^3}$ , 1994 .

[55]  Marston D. E. Conder,et al.  Determination of all Regular Maps of Small Genus , 2001, J. Comb. Theory, Ser. B.

[56]  F. A. Sherk.  The regular maps on a surface of genus three , 1959 .

[57]  Jürgen Richter-Gebert.  Realization Spaces of Polytopes , 1996 .

[58]  Bhaskar Bagchi,et al.  Combinatorial triangulations of homology spheres , 2005, Discret. Math..

[59]  Jörg M. Wills,et al.  Regular polyhedra with hidden symmetries , 1988 .

[60]  Aleksandar Mijatovic.  Simplifying triangulations of S^3 , 2000 .

[61]  Sergei Matveev,et al.  Algorithmic Topology and Classification of 3-Manifolds , 2003 .

[62]  Amos Altshuler,et al.  A non-Schlegelian polyhedral map on the torus , 1984 .

[63]  Simon A. King.  How to make a triangulation of S^3 polytopal , 2000 .

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

[65]  Frank H. Lutz,et al.  Simplicial Manifolds, Bistellar Flips and a 16-Vertex Triangulation of the Poincaré Homology 3-Sphere , 2000, Exp. Math..

[67]  Frank H. Lutz,et al.  Konrad-zuse-zentrum F ¨ Ur Informationstechnik Berlin Triangulated Manifolds with Few Vertices: Combinatorial Manifolds , 2022 .

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

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

[70]  G. Perelman.  Finite extinction time for the solutions to the Ricci flow on certain three-manifolds , 2003, math/0307245.

[71]  D. Epstein,et al.  Three-dimensional manifolds , 1960 .

[72]  Jörg M. Wills,et al.  Handbook of Convex Geometry , 1993 .

[73]  Egon Schulte,et al.  Polyhedral Maps , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

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

[75]  G. Ziegler,et al.  Polyhedral surfaces in wedge products , 2009, 0908.3159.

[76]  Walther Dyck.  Notiz über eine reguläre Riemann'sche Fläche vom Geschlechte drei und die zugehörige “Normalcurve” vierter Ordnung , 1880 .

[77]  W. Thurston.  Three dimensional manifolds, Kleinian groups and hyperbolic geometry , 1982 .

[78]  Amos Altshuler.  Manifolds in stacked 4-polytopes , 1971 .

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

[80]  S. A. Lavrenchenko,et al.  Irreducible triangulations of the torus , 1990 .

[81]  H. Coxeter,et al.  Generators and relations for discrete groups , 1957 .

[82]  M. Ellingham,et al.  Triangular embeddings of complete graphs (neighborly maps) with 12 and 13 vertices , 2005 .

[83]  Sergei Matveev,et al.  Algorithmic Topology and Classification of 3-Manifolds (Algorithms and Computation in Mathematics) , 2007 .

[84]  G. Ringel.  Map Color Theorem , 1974 .

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

[86]  David W. Barnette.  The Triangulations of the 3-Sphere with up to 8 Vertices , 1973, J. Comb. Theory, Ser. A.

[87]  Abigail Thompson,et al.  THIN POSITION AND THE RECOGNITION PROBLEM FOR S3 , 1994 .

[88]  B. McKay,et al.  Fast generation of planar graphs , 2007 .

[89]  J. Matousek,et al.  Using The Borsuk-Ulam Theorem , 2007 .

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

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

[92]  Afra Zomorodian,et al.  Computational topology , 2010 .

[93]  John Stillwell,et al.  Three-Dimensional Manifolds , 1980 .

[94]  G. Ziegler.  Polyhedral Surfaces of High Genus , 2004, math/0412093.

[95]  Sergei Matveev,et al.  An algorithm for the recognition of 3-spheres (according to Thompson) , 1995 .

[96]  Frank H. Lutz,et al.  Konrad-zuse-zentrum F ¨ Ur Informationstechnik Berlin Enumeration and Random Realization of Triangulated Surfaces Enumeration and Random Realization of Triangulated Surfaces , 2022 .

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

[98]  Jiirgen Bokowski.  A geometric realization without self-intersections does exist for Dyck's regular map , 1989, Discret. Comput. Geom..

[99]  Kenneth H. Rosen,et al.  Discrete Mathematics and its applications , 2000 .

[100]  Göran Schild,et al.  Some minimal nonembeddable complexes , 1993 .

[101]  U. Brehm.  Maximally symmetric polyhedral realizations of Dyck's regular map , 1987 .

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

[103]  E. Schulte,et al.  A Polyhedral Realization of Felix Klein's Map {3, 7}8 on a Riemann Surface of Genus 3 , 1985 .

[104]  D. Walkup.  The lower bound conjecture for 3- and 4-manifolds , 1970 .

[105]  Ulrich Brehm,et al.  Realizability of the torus and the projective plane in ℝ4 , 1995 .

[106]  B. Everitt.  The Cambridge Dictionary of Statistics , 1998 .

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