5,624 research outputs found
Expansive Motions and the Polytope of Pointed Pseudo-Triangulations
We introduce the polytope of pointed pseudo-triangulations of a point set in
the plane, defined as the polytope of infinitesimal expansive motions of the
points subject to certain constraints on the increase of their distances. Its
1-skeleton is the graph whose vertices are the pointed pseudo-triangulations of
the point set and whose edges are flips of interior pseudo-triangulation edges.
For points in convex position we obtain a new realization of the
associahedron, i.e., a geometric representation of the set of triangulations of
an n-gon, or of the set of binary trees on n vertices, or of many other
combinatorial objects that are counted by the Catalan numbers. By considering
the 1-dimensional version of the polytope of constrained expansive motions we
obtain a second distinct realization of the associahedron as a perturbation of
the positive cell in a Coxeter arrangement.
Our methods produce as a by-product a new proof that every simple polygon or
polygonal arc in the plane has expansive motions, a key step in the proofs of
the Carpenter's Rule Theorem by Connelly, Demaine and Rote (2000) and by
Streinu (2000).Comment: 40 pages, 7 figures. Changes from v1: added some comments (specially
to the "Further remarks" in Section 5) + changed to final book format. This
version is to appear in "Discrete and Computational Geometry -- The
Goodman-Pollack Festschrift" (B. Aronov, S. Basu, J. Pach, M. Sharir, eds),
series "Algorithms and Combinatorics", Springer Verlag, Berli
Faces of Birkhoff Polytopes
The Birkhoff polytope B(n) is the convex hull of all (n x n) permutation
matrices, i.e., matrices where precisely one entry in each row and column is
one, and zeros at all other places. This is a widely studied polytope with
various applications throughout mathematics.
In this paper we study combinatorial types L of faces of a Birkhoff polytope.
The Birkhoff dimension bd(L) of L is the smallest n such that B(n) has a face
with combinatorial type L.
By a result of Billera and Sarangarajan, a combinatorial type L of a
d-dimensional face appears in some B(k) for k less or equal to 2d, so bd(L) is
at most d. We will characterize those types whose Birkhoff dimension is at
least 2d-3, and we prove that any type whose Birkhoff dimension is at least d
is either a product or a wedge over some lower dimensional face. Further, we
computationally classify all d-dimensional combinatorial types for d between 2
and 8.Comment: 29 page
Positive Geometries and Differential Forms with Non-Logarithmic Singularities I
Positive geometries encode the physics of scattering amplitudes in flat
space-time and the wavefunction of the universe in cosmology for a large class
of models. Their unique canonical forms, providing such quantum mechanical
observables, are characterised by having only logarithmic singularities along
all the boundaries of the positive geometry. However, physical observables have
logarithmic singularities just for a subset of theories. Thus, it becomes
crucial to understand whether a similar paradigm can underlie their structure
in more general cases. In this paper we start a systematic investigation of a
geometric-combinatorial characterisation of differential forms with
non-logarithmic singularities, focusing on projective polytopes and related
meromorphic forms with multiple poles. We introduce the notions of covariant
forms and covariant pairings. Covariant forms have poles only along the
boundaries of the given polytope; moreover, their leading Laurent coefficients
along any of the boundaries are still covariant forms on the specific boundary.
Whereas meromorphic forms in covariant pairing with a polytope are associated
to a specific (signed) triangulation, in which poles on spurious boundaries do
not cancel completely, but their order is lowered. These meromorphic forms can
be fully characterised if the polytope they are associated to is viewed as the
restriction of a higher dimensional one onto a hyperplane. The canonical form
of the latter can be mapped into a covariant form or a form in covariant
pairing via a covariant restriction. We show how the geometry of the higher
dimensional polytope determines the structure of these differential forms.
Finally, we discuss how these notions are related to Jeffrey-Kirwan residues
and cosmological polytopes.Comment: 47 pages, figures in Tik
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