710 research outputs found
Tight triangulations of closed 3-manifolds
It is well known that a triangulation of a closed 2-manifold is tight with
respect to a field of characteristic two if and only if it is neighbourly; and
it is tight with respect to a field of odd characteristic if and only if it is
neighbourly and orientable. No such characterization of tightness was
previously known for higher dimensional manifolds. In this paper, we prove that
a triangulation of a closed 3-manifold is tight with respect to a field of odd
characteristic if and only if it is neighbourly, orientable and stacked. In
consequence, the K\"{u}hnel-Lutz conjecture is valid in dimension three for
fields of odd characteristic.
Next let be a field of characteristic two. It is known that, in
this case, any neighbourly and stacked triangulation of a closed 3-manifold is
-tight. For triangulated closed 3-manifolds with at most 71
vertices or with first Betti number at most 188, we show that the converse is
true. But the possibility of an -tight non-stacked triangulation on
a larger number of vertices remains open. We prove the following upper bound
theorem on such triangulations. If an -tight triangulation of a
closed 3-manifold has vertices and first Betti number , then
. Equality holds here if and only if all
the vertex links of the triangulation are connected sums of boundary complexes
of icosahedra.Comment: 21 pages, 1 figur
On stacked triangulated manifolds
We prove two results on stacked triangulated manifolds in this paper: (a)
every stacked triangulation of a connected manifold with or without boundary is
obtained from a simplex or the boundary of a simplex by certain combinatorial
operations; (b) in dimension , if is a tight connected
closed homology -manifold whose th homology vanishes for ,
then is a stacked triangulation of a manifold.These results give
affirmative answers to questions posed by Novik and Swartz and by Effenberger.Comment: 11 pages, minor changes in the organization of the paper, add
information about recent result
The complexity of the normal surface solution space
Normal surface theory is a central tool in algorithmic three-dimensional
topology, and the enumeration of vertex normal surfaces is the computational
bottleneck in many important algorithms. However, it is not well understood how
the number of such surfaces grows in relation to the size of the underlying
triangulation. Here we address this problem in both theory and practice. In
theory, we tighten the exponential upper bound substantially; furthermore, we
construct pathological triangulations that prove an exponential bound to be
unavoidable. In practice, we undertake a comprehensive analysis of millions of
triangulations and find that in general the number of vertex normal surfaces is
remarkably small, with strong evidence that our pathological triangulations may
in fact be the worst case scenarios. This analysis is the first of its kind,
and the striking behaviour that we observe has important implications for the
feasibility of topological algorithms in three dimensions.Comment: Extended abstract (i.e., conference-style), 14 pages, 8 figures, 2
tables; v2: added minor clarification
Stacked polytopes and tight triangulations of manifolds
Tightness of a triangulated manifold is a topological condition, roughly
meaning that any simplexwise linear embedding of the triangulation into
euclidean space is "as convex as possible". It can thus be understood as a
generalization of the concept of convexity. In even dimensions,
super-neighborliness is known to be a purely combinatorial condition which
implies the tightness of a triangulation.
Here we present other sufficient and purely combinatorial conditions which
can be applied to the odd-dimensional case as well. One of the conditions is
that all vertex links are stacked spheres, which implies that the triangulation
is in Walkup's class . We show that in any dimension
\emph{tight-neighborly} triangulations as defined by Lutz, Sulanke and Swartz
are tight.
Furthermore, triangulations with -stacked vertex links and the centrally
symmetric case are discussed.Comment: 28 pages, 2 figure
Hamiltonian submanifolds of regular polytopes
We investigate polyhedral -manifolds as subcomplexes of the boundary
complex of a regular polytope. We call such a subcomplex {\it -Hamiltonian}
if it contains the full -skeleton of the polytope. Since the case of the
cube is well known and since the case of a simplex was also previously studied
(these are so-called {\it super-neighborly triangulations}) we focus on the
case of the cross polytope and the sporadic regular 4-polytopes. By our results
the existence of 1-Hamiltonian surfaces is now decided for all regular
polytopes.
Furthermore we investigate 2-Hamiltonian 4-manifolds in the -dimensional
cross polytope. These are the "regular cases" satisfying equality in Sparla's
inequality. In particular, we present a new example with 16 vertices which is
highly symmetric with an automorphism group of order 128. Topologically it is
homeomorphic to a connected sum of 7 copies of . By this
example all regular cases of vertices with or, equivalently, all
cases of regular -polytopes with are now decided.Comment: 26 pages, 4 figure
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