No Access

FACE PAIRING GRAPHS AND 3-MANIFOLD ENUMERATION

BENJAMIN A. BURTON

Department of Mathematics and Statistics, University of Melbourne, Parkville 3010, Victoria, Australia

https://doi.org/10.1142/S0218216504003627Cited by:10 (Source: Crossref)

Abstract

The face pairing graph of a 3-manifold triangulation is a 4-valent graph denoting which tetrahedron faces are identified with which others. We present a series of properties that must be satisfied by the face pairing graph of a closed minimal ℙ²-irreducible triangulation. In addition we present constraints upon the combinatorial structure of such a triangulation that can be deduced from its face pairing graph. These results are then applied to the enumeration of closed minimal ℙ²-irreducible 3-manifold triangulations, leading to a significant improvement in the performance of the enumeration algorithm. Results are offered for both orientable and non-orientable triangulations.

Keywords:

AMSC: Primary 57N10, Primary 57M15, Secondary 05C30

References

G. Amendola and B. Martelli, Topology Appl. 133(2), 157 (2003). Crossref, Web of Science, Google Scholar
B. A. Burton, Regina (normal surface and 3-manifold topology software), http://regina.sourceforge.net/ (1999–2004) . Google Scholar
B. A. Burton, Structures of small closed non-orientable 3-manifold triangulations, preprint, math.GT/0311113 (November 2003) . Google Scholar
P. J. Callahan, M. V. Hildebrand and J. R. Weeks, Math. Comp. 68(225), 321 (1999). Crossref, Web of Science, Google Scholar
M. V. Hildebrand and J. R. Weeks, Computers and Mathematics (Springer, New York, 1989) pp. 53–59. Crossref, Google Scholar
C. D. Hodgson and J. R. Weeks, Experiment. Math. 3(4), 261 (1994). Crossref, Google Scholar
W. Jaco, D. Letscher and J. H. Rubinstein, Topology and Geometry: Commemorating SISTAG, Contemporary Mathematics, No. 314 (American Mathematical Society, Providence, RI, 2002) pp. 107–124. Crossref, Google Scholar
W. Jaco and J. H. Rubinstein, 0-efficient triangulations of 3-manifolds, to appear in J. Differential Geom., math.GT/0207158 (July 2002) . Google Scholar
W. Jaco and J. H. Rubinstein, Layered triangulations of lens spaces, in preparation (2003) . Google Scholar
W. Jaco and J. L. Tollefson, Illinois J. Math. 39(3), 358 (1995). Crossref, Web of Science, Google Scholar
B. Martelli and C. Petronio, Experiment. Math. 10(2), 207 (2001). Crossref, Web of Science, Google Scholar
B. Martelli and C. Petronio, Illinois J. Math. 46, 755 (2002). Crossref, Web of Science, Google Scholar
B. Martelli and C. Petronio, Complexity of geometric three-manifolds, preprint, math.GT/0303249 (March 2003) . Google Scholar
S. V. Matveev, Tables of 3-manifolds up to complexity 6, Max-Planck-Institut für Mathematik Preprint Series (1998), No. 67, http://www.mpim-bonn.mpg.de/html/preprints/preprints.html . Google Scholar
J. H. Rubinstein, Proc. Int. Congress of Mathematicians (Zürich, 1994) 1 (Birkhüuser, 1995) pp. 601–611. Crossref, Google Scholar
J. H. Rubinstein, Geometric Topology (Athens, GA, 1993), AMS/IP Studies in Advanced Mathematics 2 (American Mathematical Society, 1997) pp. 1–20. Google Scholar