541 research outputs found
The infinite cyclohedron and its automorphism group
Cyclohedra are a well-known infinite familiy of finite-dimensional polytopes
that can be constructed from centrally symmetric triangulations of even-sided
polygons. In this article we introduce an infinite-dimensional analogue and
prove that the group of symmetries of our construction is a semidirect product
of a degree 2 central extension of Thompson's infinite finitely presented
simple group T with the cyclic group of order 2. These results are inspired by
a similar recent analysis by the first author of the automorphism group of an
infinite-dimensional associahedron.Comment: 18 pages, 8 figure
Asymptotically maximal families of hypersurfaces in toric varieties
A real algebraic variety is maximal (with respect to the Smith-Thom
inequality) if the sum of the Betti numbers (with coefficients)
of the real part of the variety is equal to the sum of Betti numbers of its
complex part. We prove that there exist polytopes that are not Newton polytopes
of any maximal hypersurface in the corresponding toric variety. On the other
hand we show that for any polytope there are families of hypersurfaces
with the Newton polytopes that are
asymptotically maximal when tends to infinity. We also show that
these results generalize to complete intersections.Comment: 18 pages, 1 figur
Happy endings for flip graphs
We show that the triangulations of a finite point set form a flip graph that
can be embedded isometrically into a hypercube, if and only if the point set
has no empty convex pentagon. Point sets of this type include convex subsets of
lattices, points on two lines, and several other infinite families. As a
consequence, flip distance in such point sets can be computed efficiently.Comment: 26 pages, 15 figures. Revised and expanded for journal publicatio
Lower bounds for the simplexity of the n-cube
In this paper we prove a new asymptotic lower bound for the minimal number of
simplices in simplicial dissections of -dimensional cubes. In particular we
show that the number of simplices in dissections of -cubes without
additional vertices is at least .Comment: 10 page
Moduli of Tropical Plane Curves
We study the moduli space of metric graphs that arise from tropical plane
curves. There are far fewer such graphs than tropicalizations of classical
plane curves. For fixed genus , our moduli space is a stacky fan whose cones
are indexed by regular unimodular triangulations of Newton polygons with
interior lattice points. It has dimension unless or .
We compute these spaces explicitly for .Comment: 31 pages, 25 figure
The Complexity of Finding Small Triangulations of Convex 3-Polytopes
The problem of finding a triangulation of a convex three-dimensional polytope
with few tetrahedra is proved to be NP-hard. We discuss other related
complexity results.Comment: 37 pages. An earlier version containing the sketch of the proof
appeared at the proceedings of SODA 200
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