Structures of small closed non-orientable 3-manifold triangulations

A census is presented of all closed non-orientable 3-manifold triangulations formed from at most seven tetrahedra satisfying the additional constraints of minimality and ℙ2-irreducibility. The eight different 3-manifolds represented by these 41 different triangulations are identified and described in detail, with particular attention paid to the recurring combinatorial structures that are shared amongst the different triangulations. Using these recurring structures, the resulting triangulations are generalised to infinite families that allow similar triangulations of additional 3-manifolds to be formed.

[1]  Geoffrey Hemion.  The Classification of Knots and 3-Dimensional Spaces , 1992 .

[2]  Udo Pachner,et al.  P.L. Homeomorphic Manifolds are Equivalent by Elementary 5hellingst , 1991, Eur. J. Comb..

[3]  Jeffrey R. Weeks,et al.  A census of cusped hyperbolic 3-manifolds , 1999, Math. Comput..

[4]  J. Hyam Rubinstein,et al.  0-Efficient Triangulations of 3-Manifolds , 2002 .

[5]  David Letscher,et al.  Algorithms for essential surfaces in 3-manifolds , 2002 .

[6]  Bruno Martelli,et al.  Three-Manifolds Having Complexity At Most 9 , 2001, Exp. Math..

[7]  Horst Schubert,et al.  Bestimmung der Primfaktorzerlegung von Verkettungen , 1961 .

[8]  Erich Kaltofen,et al.  Computers and Mathematics , 1989, Springer US.

[9]  Benjamin A. Burton.  MINIMAL TRIANGULATIONS AND NORMAL SURFACES , 2003 .

[10]  W. Harvey.  THE CLASSIFICATION OF KNOTS AND 3‐DIMENSIONAL SPACES , 1995 .

[11]  Benjamin A. Burton.  Introducing Regina, The 3-Manifold Topology Software , 2004, Exp. Math..

[12]  R. Smullyan.  ANNALS OF MATHEMATICS STUDIES , 1961 .

[13]  Benjamin A. Burton.  FACE PAIRING GRAPHS AND 3-MANIFOLD ENUMERATION , 2004 .

[14]  Bruno Martelli.  Complexity of 3-manifolds , 2004 .

[15]  Bruno Martelli,et al.  Complexity of Geometric Three-manifolds , 2003 .

[16]  Vladimir Turaev,et al.  State sum invariants of 3 manifolds and quantum 6j symbols , 1992 .

[17]  Gennaro Amendola,et al.  Non-orientable 3-manifolds of complexity up to 7 , 2003 .

[18]  Gennaro Amendola,et al.  Non-orientable 3-manifolds of small complexity , 2002 .

[19]  Martin Hildebrand,et al.  A Computer Generated Census of Cusped Hyperbolic 3-Manifolds , 1989, Computers and Mathematics.

[20]  Hellmuth Kneser,et al.  Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten. , 1929 .

[21]  William Jaco,et al.  Algorithms for the complete decomposition of a closed $3$-manifold , 1995 .