771 research outputs found
On k-stacked polytopes
AbstractIt is proved that equality in the Generalized Simplicial Lower Bound Conjecture can always be obtained by k-stacked polytopes
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
On the generalized lower bound conjecture for polytopes and spheres
In 1971, McMullen and Walkup posed the following conjecture, which is called
the generalized lower bound conjecture: If is a simplicial -polytope
then its -vector satisfies . Moreover, if for some then can be triangulated without introducing simplices of
dimension .
The first part of the conjecture was solved by Stanley in 1980 using the hard
Lefschetz theorem for projective toric varieties. In this paper, we give a
proof of the remaining part of the conjecture. In addition, we generalize this
property to a certain class of simplicial spheres, namely those admitting the
weak Lefschetz property.Comment: 14 pages, improved presentatio
Embedding Stacked Polytopes on a Polynomial-Size Grid
A stacking operation adds a -simplex on top of a facet of a simplicial
-polytope while maintaining the convexity of the polytope. A stacked
-polytope is a polytope that is obtained from a -simplex and a series of
stacking operations. We show that for a fixed every stacked -polytope
with vertices can be realized with nonnegative integer coordinates. The
coordinates are bounded by , except for one axis, where the
coordinates are bounded by . The described realization can be
computed with an easy algorithm.
The realization of the polytopes is obtained with a lifting technique which
produces an embedding on a large grid. We establish a rounding scheme that
places the vertices on a sparser grid, while maintaining the convexity of the
embedding.Comment: 22 pages, 10 Figure
On the prevalence of elliptic and genus one fibrations among toric hypersurface Calabi-Yau threefolds
We systematically analyze the fibration structure of toric hypersurface
Calabi-Yau threefolds with large and small Hodge numbers. We show that there
are only four such Calabi-Yau threefolds with or that do not have manifest elliptic or genus one fibers arising from a
fibration of the associated 4D polytope. There is a genus one fibration
whenever either Hodge number is 150 or greater, and an elliptic fibration when
either Hodge number is 228 or greater. We find that for small the
fraction of polytopes in the KS database that do not have a genus one or
elliptic fibration drops exponentially as increases. We also consider
the different toric fiber types that arise in the polytopes of elliptic
Calabi-Yau threefolds.Comment: 37 pages, 8 figures; v2: references adde
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