2,860 research outputs found
Irreducible Triangulations are Small
A triangulation of a surface is \emph{irreducible} if there is no edge whose
contraction produces another triangulation of the surface. We prove that every
irreducible triangulation of a surface with Euler genus has at most
vertices. The best previous bound was .Comment: v2: Referees' comments incorporate
Face pairing graphs and 3-manifold enumeration
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 P^2-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 P^2-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.Comment: 30 pages, 57 figures; v2: clarified some passages and generalised the
final theorem to the non-orientable case; v3: fixed a flaw in the proof of
the conical face lemm
Enumeration of non-orientable 3-manifolds using face pairing graphs and union-find
Drawing together techniques from combinatorics and computer science, we
improve the census algorithm for enumerating closed minimal P^2-irreducible
3-manifold triangulations. In particular, new constraints are proven for face
pairing graphs, and pruning techniques are improved using a modification of the
union-find algorithm. Using these results we catalogue all 136 closed
non-orientable P^2-irreducible 3-manifolds that can be formed from at most ten
tetrahedra.Comment: 37 pages, 34 figure
Irreducible triangulations of surfaces with boundary
A triangulation of a surface is irreducible if no edge can be contracted to
produce a triangulation of the same surface. In this paper, we investigate
irreducible triangulations of surfaces with boundary. We prove that the number
of vertices of an irreducible triangulation of a (possibly non-orientable)
surface of genus g>=0 with b>=0 boundaries is O(g+b). So far, the result was
known only for surfaces without boundary (b=0). While our technique yields a
worse constant in the O(.) notation, the present proof is elementary, and
simpler than the previous ones in the case of surfaces without boundary
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