14 research outputs found
Associahedra via spines
An associahedron is a polytope whose vertices correspond to triangulations of
a convex polygon and whose edges correspond to flips between them. Using
labeled polygons, C. Hohlweg and C. Lange constructed various realizations of
the associahedron with relevant properties related to the symmetric group and
the classical permutahedron. We introduce the spine of a triangulation as its
dual tree together with a labeling and an orientation. This notion extends the
classical understanding of the associahedron via binary trees, introduces a new
perspective on C. Hohlweg and C. Lange's construction closer to J.-L. Loday's
original approach, and sheds light upon the combinatorial and geometric
properties of the resulting realizations of the associahedron. It also leads to
noteworthy proofs which shorten and simplify previous approaches.Comment: 27 pages, 11 figures. Version 5: minor correction
Polytopality and Cartesian products of graphs
We study the question of polytopality of graphs: when is a given graph the
graph of a polytope? We first review the known necessary conditions for a graph
to be polytopal, and we provide several families of graphs which satisfy all
these conditions, but which nonetheless are not graphs of polytopes. Our main
contribution concerns the polytopality of Cartesian products of non-polytopal
graphs. On the one hand, we show that products of simple polytopes are the only
simple polytopes whose graph is a product. On the other hand, we provide a
general method to construct (non-simple) polytopal products whose factors are
not polytopal.Comment: 21 pages, 10 figure
Multi-triangulations as complexes of star polygons
Maximal -crossing-free graphs on a planar point set in convex
position, that is, -triangulations, have received attention in recent
literature, with motivation coming from several interpretations of them.
We introduce a new way of looking at -triangulations, namely as complexes
of star polygons. With this tool we give new, direct, proofs of the fundamental
properties of -triangulations, as well as some new results. This
interpretation also opens-up new avenues of research, that we briefly explore
in the last section.Comment: 40 pages, 24 figures; added references, update Section
Subword complexes, cluster complexes, and generalized multi-associahedra
In this paper, we use subword complexes to provide a uniform approach to
finite type cluster complexes and multi-associahedra. We introduce, for any
finite Coxeter group and any nonnegative integer k, a spherical subword complex
called multi-cluster complex. For k=1, we show that this subword complex is
isomorphic to the cluster complex of the given type. We show that multi-cluster
complexes of types A and B coincide with known simplicial complexes, namely
with the simplicial complexes of multi-triangulations and centrally symmetric
multi-triangulations respectively. Furthermore, we show that the multi-cluster
complex is universal in the sense that every spherical subword complex can be
realized as a link of a face of the multi-cluster complex.Comment: 26 pages, 3 Tables, 2 Figures; final versio
Many non-equivalent realizations of the associahedron
Hohlweg and Lange (2007) and Santos (2004, unpublished) have found two
different ways of constructing exponential families of realizations of the
n-dimensional associahedron with normal vectors in {0,1,-1}^n, generalizing the
constructions of Loday (2004) and Chapoton-Fomin-Zelevinsky (2002). We classify
the associahedra obtained by these constructions modulo linear equivalence of
their normal fans and show, in particular, that the only realization that can
be obtained with both methods is the Chapoton-Fomin-Zelevinsky (2002)
associahedron.
For the Hohlweg-Lange associahedra our classification is a priori coarser
than the classification up to isometry of normal fans, by
Bergeron-Hohlweg-Lange-Thomas (2009). However, both yield the same classes. As
a consequence, we get that two Hohlweg-Lange associahedra have linearly
equivalent normal fans if and only if they are isometric.
The Santos construction, which produces an even larger family of
associahedra, appears here in print for the first time. Apart of describing it
in detail we relate it with the c-cluster complexes and the denominator fans in
cluster algebras of type A.
A third classical construction of the associahedron, as the secondary
polytope of a convex n-gon (Gelfand-Kapranov-Zelevinsky, 1990), is shown to
never produce a normal fan linearly equivalent to any of the other two
constructions.Comment: 30 pages, 13 figure
Computing pseudotriangulations via branched coverings
We describe an efficient algorithm to compute a pseudotriangulation of a
finite planar family of pairwise disjoint convex bodies presented by its
chirotope. The design of the algorithm relies on a deepening of the theory of
visibility complexes and on the extension of that theory to the setting of
branched coverings. The problem of computing a pseudotriangulation that
contains a given set of bitangent line segments is also examined.Comment: 66 pages, 39 figure
El-labelings and canonical spanning trees for subword complexes
Abstract. We describe edge labelings of the increasing flip graph of a subword complex on a finite Coxeter group, and study applications thereof. On the one hand, we show that they provide canonical spanning trees of the facet-ridge graph of the subword complex, describe inductively these trees, and present their close relations to greedy facets. Searching these trees yields an efficient algorithm to generate all facets of the subword complex, which extends the greedy flip algorithm for pointed pseudotriangulations. On the other hand, when the increasing flip graph is a Hasse diagram, we show that the edge labeling is indeed an EL-labeling and derive further combinatorial properties of paths in the increasing flip graph. These results apply in particular to Cambrian lattices, in which case a similar EL-labeling was recently studied by M. Kallipoliti and H. Mühle. Content
Distributing Reactive Systems
This paper addresses the problem of distributing reactive systems. We first show that the use of synchronous languages allows a natural parallel description of such systems, regardless of any distribution problems. Then, a desired distribution can be easily specified, and achieved with the algorithm presented here. This distribution technique provides distributed programs with the same safety, test and debug facilities as ordinary sequential programs. Finally, the implementation of such distributed programs only requires a very simple communication protocol ("first in first out" queues), thereby reducing the need for large distributed real-time executives