577 research outputs found
Dyck path triangulations and extendability
We introduce the Dyck path triangulation of the cartesian product of two
simplices . The maximal simplices of this
triangulation are given by Dyck paths, and its construction naturally
generalizes to produce triangulations of
using rational Dyck paths. Our study of the Dyck path triangulation is
motivated by extendability problems of partial triangulations of products of
two simplices. We show that whenever , any triangulation of
extends to a unique triangulation of
. Moreover, with an explicit construction, we
prove that the bound is optimal. We also exhibit interesting
interpretations of our results in the language of tropical oriented matroids,
which are analogous to classical results in oriented matroid theory.Comment: 15 pages, 14 figures. Comments very welcome
Valuations for matroid polytope subdivisions
We prove that the ranks of the subsets and the activities of the bases of a
matroid define valuations for the subdivisions of a matroid polytope into
smaller matroid polytopes.Comment: 19 pages. 2 figures; added section 6 + other correction
Splitting Polytopes
A split of a polytope is a (regular) subdivision with exactly two maximal
cells. It turns out that each weight function on the vertices of admits a
unique decomposition as a linear combination of weight functions corresponding
to the splits of (with a split prime remainder). This generalizes a result
of Bandelt and Dress [Adv. Math. 92 (1992)] on the decomposition of finite
metric spaces.
Introducing the concept of compatibility of splits gives rise to a finite
simplicial complex associated with any polytope , the split complex of .
Complete descriptions of the split complexes of all hypersimplices are
obtained. Moreover, it is shown that these complexes arise as subcomplexes of
the tropical (pre-)Grassmannians of Speyer and Sturmfels [Adv. Geom. 4 (2004)].Comment: 25 pages, 7 figures; minor corrections and change
Dyck path triangulations and extendability (extended abstract)
International audienceWe introduce the Dyck path triangulation of the cartesian product of two simplices . The maximal simplices of this triangulation are given by Dyck paths, and its construction naturally generalizes to produce triangulations of using rational Dyck paths. Our study of the Dyck path triangulation is motivated by extendability problems of partial triangulations of products of two simplices. We show that whenever, any triangulations of extends to a unique triangulation of . Moreover, with an explicit construction, we prove that the bound is optimal. We also exhibit interpretations of our results in the language of tropical oriented matroids, which are analogous to classical results in oriented matroid theory.Nous introduisons la triangulation par chemins de Dyck du produit cartésien de deux simplexes . Les simplexes maximaux de cette triangulation sont donnés par des chemins de Dyck, et cette construction se généralise de façon naturelle pour produire des triangulations qui utilisent des chemins de Dyck rationnels. Notre étude de la triangulation par chemins de Dyck est motivée par des problèmes de prolongement de triangulations partielles de produits de deux simplexes. On montre que alors toute triangulation de se prolonge en une unique triangulation de . De plus, avec une construction explicite, nous montrons que la borne est optimale. Nous présentons aussi des interprétations de nos résultats dans le langage des matroïdes orientés tropicaux, qui sont analogues aux résultats classiques de la théorie des matroïdes orientés
Tropical Convexity
The notions of convexity and convex polytopes are introduced in the setting
of tropical geometry. Combinatorial types of tropical polytopes are shown to be
in bijection with regular triangulations of products of two simplices.
Applications to phylogenetic trees are discussed.
Theorem 29 and Corollary 30 in the paper, relating tropical polytopes to
injective hulls, are incorrect. See the erratum at
http://www.math.uiuc.edu/documenta/vol-09/vol-09-eng.html .Comment: 20 pages, 6 figure
Cambrian triangulations and their tropical realizations
This paper develops a Cambrian extension of the work of C. Ceballos, A.
Padrol and C. Sarmiento on -Tamari lattices and their tropical
realizations. For any signature , we consider a
family of -trees in bijection with the triangulations of the
-polygon. These -trees define a flag regular
triangulation of the subpolytope of the product of simplices . The oriented
dual graph of the triangulation is the Hasse diagram
of the (type ) -Cambrian lattice of N. Reading. For any
and , we consider the restriction
of the triangulation
to the face . Its dual graph is naturally interpreted as the increasing
flip graph on certain -trees, which is shown
to be a lattice generalizing in particular the -Tamari lattices in the
Cambrian setting. Finally, we present an alternative geometric realization of
as a polyhedral complex induced
by a tropical hyperplane arrangement.Comment: 16 pages, 11 figure
Planarity in Generalized Scattering Amplitudes: PK Polytope, Generalized Root Systems and Worldsheet Associahedra
In this paper we study the role of planarity in generalized scattering
amplitudes, through several closely interacting structures in combinatorics,
algebraic and tropical geometry.
The generalized biadjoint scalar amplitude, introduced recently by
Cachazo-Early-Guevara-Mizera (CEGM), is a rational function of homogeneous
degree in independent variables; its poles can
be constructed directly from the rays of the positive tropical Grassmannian.
We introduce for each pair of integers with a system
of generalized positive roots which arises as a specialization of the planar
basis of kinematic invariants. We prove that the higher root polytope
has volume the k-dimensional Catalan number
, via a flag unimodular triangulation into simplices, in
bijection with noncrossing collections of -element subsets. We also give a
bijection between certain positroidal subdivisions, called tripods, of the
hypersimplex and noncrossing pairs of 3-element subsets that are
not weakly separated.
We show that the facets of the Planar Kinematics (PK) polytope, introduced
recently by Cachazo and the author, are exactly the
generalized positive roots. We show that the PK specialization of the
generalized biadjoint amplitude evaluates to .
Looking forward, we give defining equations and conjecture explicit solutions
using via a notion of compatibility
degree for noncrossing collections, for a two parameter family of generalized
worldsheet associahedra . These specialize when to a
certain dihedrally invariant partial compactification of the configuration
space of distinct points in . Many detailed
examples are given throughout to motivate future work.Comment: 75 pages, 19 figure
Palm phytoliths of mid-elevation Andean forests
Palms are one of the most common tropical plant groups. They are widespread across lowland tropical forests, but many are found in higher altitudes have more constrained environmental ranges. The limited range of these species makes them particularly useful in paleoecological and paleoclimate reconstructions. Palms produce phytoliths, or silica structures, which are found in their vegetative parts (e.g., wood, leaves, etc.). Recent research has shown that several palms in the lowland tropical forests produce phytoliths that are diagnostic to the sub-family or genus-level. Here we characterize Andean palm phytoliths, and determine whether many of these species can also be identified by their silica structures. All of our sampled Andean palm species produced phytoliths, and we were able to characterize several previously unclassified morphotypes. Some species contained unique phytoliths that did not occur in other species, particularly Ceroxylon alpinium, which is indicative of specific climatic conditions. The differences in the morphologies of the Andean species indicate that palm phytolith analysis is particularly useful in paleoecological reconstructions. Future phytolith analyses will allow researchers to track how these palm species with limited environmental ranges have migrated up and down the Andean slopes as a result of past climatic change. The phytolith analyses can track local-scale vegetation dynamics, whereas pollen, which is commonly used in paleoecological reconstructions, reflects regional-scale vegetation change
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