2,351 research outputs found
Triangulated Manifolds with Few Vertices: Centrally Symmetric Spheres and Products of Spheres
The aim of this paper is to give a survey of the known results concerning
centrally symmetric polytopes, spheres, and manifolds. We further enumerate
nearly neighborly centrally symmetric spheres and centrally symmetric products
of spheres with dihedral or cyclic symmetry on few vertices, and we present an
infinite series of vertex-transitive nearly neighborly centrally symmetric
3-spheres.Comment: 26 pages, 8 figure
Classification of triples of lattice polytopes with a given mixed volume
We present an algorithm for the classification of triples of lattice
polytopes with a given mixed volume in dimension 3. It is known that the
classification can be reduced to the enumeration of so-called irreducible
triples, the number of which is finite for fixed . Following this algorithm,
we enumerate all irreducible triples of normalized mixed volume up to 4 that
are inclusion-maximal. This produces a classification of generic trivariate
sparse polynomial systems with up to 4 solutions in the complex torus, up to
monomial changes of variables. By a recent result of Esterov, this leads to a
description of all generic trivariate sparse polynomial systems that are
solvable by radicals
Face enumeration on simplicial complexes
Let be a closed triangulable manifold, and let be a
triangulation of . What is the smallest number of vertices that can
have? How big or small can the number of edges of be as a function of
the number of vertices? More generally, what are the possible face numbers
(-numbers, for short) that can have? In other words, what
restrictions does the topology of place on the possible -numbers of
triangulations of ?
To make things even more interesting, we can add some combinatorial
conditions on the triangulations we are considering (e.g., flagness,
balancedness, etc.) and ask what additional restrictions these combinatorial
conditions impose. While only a few theorems in this area of combinatorics were
known a couple of decades ago, in the last ten years or so, the field simply
exploded with new results and ideas. Thus we feel that a survey paper is long
overdue. As new theorems are being proved while we are typing this chapter, and
as we have only a limited number of pages, we apologize in advance to our
friends and colleagues, some of whose results will not get mentioned here.Comment: Chapter for upcoming IMA volume Recent Trends in Combinatoric
Combinatorial 3-manifolds with 10 vertices
We give a complete enumeration of all combinatorial 3-manifolds with 10
vertices: There are precisely 247882 triangulated 3-spheres with 10 vertices as
well as 518 vertex-minimal triangulations of the sphere product
and 615 triangulations of the twisted sphere product S^2_\times_S^1.
All the 3-spheres with up to 10 vertices are shellable, but there are 29
vertex-minimal non-shellable 3-balls with 9 vertices.Comment: 9 pages, minor revisions, to appear in Beitr. Algebra Geo
Complementary vertices and adjacency testing in polytopes
Our main theoretical result is that, if a simple polytope has a pair of
complementary vertices (i.e., two vertices with no facets in common), then it
has at least two such pairs, which can be chosen to be disjoint. Using this
result, we improve adjacency testing for vertices in both simple and non-simple
polytopes: given a polytope in the standard form {x \in R^n | Ax = b and x \geq
0} and a list of its V vertices, we describe an O(n) test to identify whether
any two given vertices are adjacent. For simple polytopes this test is perfect;
for non-simple polytopes it may be indeterminate, and instead acts as a filter
to identify non-adjacent pairs. Our test requires an O(n^2 V + n V^2)
precomputation, which is acceptable in settings such as all-pairs adjacency
testing. These results improve upon the more general O(nV) combinatorial and
O(n^3) algebraic adjacency tests from the literature.Comment: 14 pages, 5 figures. v1: published in COCOON 2012. v2: full journal
version, which strengthens and extends the results in Section 2 (see p1 of
the paper for details
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