176 research outputs found
Minimal flag triangulations of lower-dimensional manifolds
We prove the following results on flag triangulations of 2- and 3-manifolds.
In dimension 2, we prove that the vertex-minimal flag triangulations of
and have 11 and 12 vertices,
respectively. In general, we show that (resp. ) vertices suffice
to obtain a flag triangulation of the connected sum of copies of
(resp. ). In dimension 3, we
describe an algorithm based on the Lutz-Nevo theorem which provides supporting
computational evidence for the following generalization of the Charney-Davis
conjecture: for any flag 3-manifold, ,
where is the number of -dimensional faces and is the first
Betti number over a field. The conjecture is tight in the sense that for any
value of , there exists a flag 3-manifold for which the equality
holds.Comment: 6 figures, 3 tables, 19 pages. Final version with a few typos
correcte
Balanced triangulations on few vertices and an implementation of cross-fips
A
d
-dimensional simplicial complex is balanced if the underlying graph is
(
d
+
1
)
-colorable. We present an implementation of cross-flips, a set of local moves
introduced by Izmestiev, Klee and Novik which connect any two PL-homeomorphic
balanced combinatorial manifolds. As a result we exhibit a vertex minimal balanced
triangulation of the real projective plane, of the dunce hat and of the real projective
space, as well as several balanced triangulations of surfaces and 3-manifolds on few
vertices. In particular we construct small balanced triangulations of the 3-sphere
that are non-shellable and shellable but not vertex decomposable
Detecting genus in vertex links for the fast enumeration of 3-manifold triangulations
Enumerating all 3-manifold triangulations of a given size is a difficult but
increasingly important problem in computational topology. A key difficulty for
enumeration algorithms is that most combinatorial triangulations must be
discarded because they do not represent topological 3-manifolds. In this paper
we show how to preempt bad triangulations by detecting genus in
partially-constructed vertex links, allowing us to prune the enumeration tree
substantially.
The key idea is to manipulate the boundary edges surrounding partial vertex
links using expected logarithmic time operations. Practical testing shows the
resulting enumeration algorithm to be significantly faster, with up to 249x
speed-ups even for small problems where comparisons are feasible. We also
discuss parallelisation, and describe new data sets that have been obtained
using high-performance computing facilities.Comment: 16 pages, 7 figures, 3 tables; v2: minor revisions; to appear in
ISSAC 201
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Discrete Differential Geometry
Discrete Differential Geometry is a broad new area where differential geometry (studying smooth curves, surfaces and other manifolds) interacts with discrete geometry (studying polyhedral manifolds), using tools and ideas from all parts of mathematics. This report documents the 29 lectures at the first Oberwolfach workshop in this subject, with topics ranging from discrete integrable systems, polyhedra, circle packings and tilings to applications in computer graphics and geometry processing. It also includes a list of open problems posed at the problem session
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