Simple Polytopes Research Articles

Manifolds of isospectral arrow matrices

An arrow matrix is a matrix with zeros outside the main diagonal, the first row and the first column. We consider the space of Hermitian arrow -matrices with fixed simple spectrum . We prove that this space is a smooth -manifold with a locally standard torus action: we describe the topology and combinatorics of its orbit space. If , the orbit space is not a polytope, hence is not a quasitoric manifold. However, there is an action of a semidirect product on , and the orbit space of this action is a certain simple polytope obtained from the cube by cutting off codimension-2 faces. In the case , the space is a solid torus with boundary subdivided into hexagons in a regular way. This description allows us to compute the cohomology ring and equivariant cohomology ring of the 6-dimensional manifold and another manifold, its twin. Bibliography: 32 titles.

Gamma-positivity of derangement polynomials and binomial Eulerian polynomials for colored permutations

The binomial Eulerian polynomials, first introduced in work of Postnikov, Reiner and Williams, are γ-positive polynomials and can be interpreted as h-polynomials of certain flag simplicial polytopes. Recently, Athanasiadis studied analogs of these polynomials for colored permutations and proved that they can be written as the sums of two γ-positive polynomials. In this paper, we find combinatorial interpretations of Athanasiadis' γ-positive polynomials, which leads to an alternative proof of their γ-positivity expansions using the method of group actions on colored permutations. Two results are presented. The first one is to give the γ-coefficients of the symmetric decompositions of binomial Eulerian polynomials for colored permutations, which answers a problem of Athanasiadis (2020) [3]. The second one is to give a direct combinatorial proof on the γ-expansions in the symmetric decompositions of colored derangement polynomials, which answers another problem asked by Athanasiadis (2018) [2].

Scattering amplitudes and simple canonical forms for simple polytopes

We provide an efficient recursive formula to compute the canonical forms of arbitrary d-dimensional simple polytopes, which are convex polytopes such that every vertex lies precisely on d facets. For illustration purposes, we explicitly derive recursive formulae for the canonical forms of Stokes polytopes, which play a similar role for a theory with quartic interaction as the Associahedron does in planar bi-adjoint ϕ3 theory. As a by-product, our formula also suggests a new way to obtain the full planar amplitude in ϕ4 theory by taking suitable limits of the canonical forms of constituent Stokes polytopes.

ON THE DIMENSIONS OF THE REALIZATION SPACES OF POLYTOPES

Robertson (1988) suggested a model for the realization space of a convex d-dimensional polytope and an approach via the implicit function theorem, which -- in the case of a full rank Jacobian -- proves that the realization space is a manifold of dimension NG(P):=d(f_0+f_{d-1})-f_{0,d-1}, which is the natural guess for the dimension given by the number of variables minus the number of quadratic equations that are used in the definition of the realization space. While this indeed holds for many natural classes of polytopes (including simple and simplicial polytopes, as well as all polytopes of dimension at most 3),and Robertson claimed this to be true for all polytopes, Mnev's (1986/1988) Universality Theorem implies that it is not true in general: Indeed, (1) the centered realization space is not a smoothly embedded manifold in general, and (2) it does not have the dimension NG(P) in general. In this paper we develop Jacobian criteria for the analysis of realization spaces. From these we get easily that for various large and natural classes of polytopes the realization spaces are indeed manifolds, whose dimensions are given by NG(P). However, we also identify the smallest polytopes where the dimension count (2) and thus Robertson's claim fails, among them the bipyramid over a triangular prism. For the property (1), we analyze the classical 24-cell: We show that the realization space has at least the dimension 48, and it has points where it is a manifold of this dimension, but it is not smoothly embedded as a manifold everywhere.

HOMOTOPY CONTINUATION FOR THE SPECTRA OF PERSISTENT LAPLACIANS.

The p-persistent q-combinatorial Laplacian defined for a pair of simplicial complexes is a generalization of the q-combinatorial Laplacian. Given a filtration, the spectra of persistent combinatorial Laplacians not only recover the persistent Betti numbers of persistent homology but also provide extra multiscale geometrical information of the data. Paired with machine learning algorithms, the persistent Laplacian has many potential applications in data science. Seeking different ways to find the spectrum of an operator is an active research topic, becoming interesting when ideas are originated from multiple fields. In this work, we explore an alternative approach for the spectrum of persistent Laplacians. As the eigenvalues of a persistent Laplacian matrix are the roots of its characteristic polynomial, one may attempt to find the roots of the characteristic polynomial by homotopy continuation, and thus resolving the spectrum of the corresponding persistent Laplacian. We consider a set of simple polytopes and small molecules to prove the principle that algebraic topology, combinatorial graph, and algebraic geometry can be integrated to understand the shape of data.

GKM theory for orbifold stratified spaces and application to singular toric varieties

We study the GKM theory for a equivariant stratified space having orbifold structures in its successive quotients. Then, we introduce the notion of an almost simple polytope, as well as a divisive toric variety generalizing the concept of a divisive weighted projective space. We employ the GKM theory to compute the generalized equivariant cohomology theories of toric varieties associated to almost simple polytopes and divisive toric varieties.

A CAD-free methodology for volume and mass properties computation of 3-D lifting surfaces and wing-box structures

The geometry, volume, and mass properties (GVM) of lifting surfaces and wing box structures play an important role in aircraft design and optimization. Commercial Computer-Aided-Design packages can be employed to determine these features; however, they are difficult to embed in multidisciplinary frameworks because of their limited scripting capabilities and their black-box structure. In this context, the present work introduces an open-source, fully scriptable, high fidelity, and low computational demanding methodology to compute the volume and mass properties of lifting surfaces and wingbox structures using their three-dimensional geometric definition. NURBS modeling is employed to generate the 3-D geometry and derive the vertex representation of the lifting surface. The volume computation is based on the Divergence Theorem and employs a structured triangulation approach, tailored for lifting surfaces and their cross sections. The mass properties (i.e. center of mass and moments of inertia) are calculated as a system of mass particles. For this, a mass distribution model, based on the thickness and chord distribution throughout the lifting surface, has been elaborated. The methodology for volume computation and the mass distribution model has been validated analytically using a simple polytope that resembles to a lifting surface. The convergence and the robustness of the methodology has been evaluated for a straight tapered lifting surface. The results indicate a maximum error, compared with a commercial CAD software, of 0.3% and 0.5% for the volume and mass properties computation, respectively. In addition, to assess its suitability for complex lifting surfaces, the NASA Common Research Model aircraft (NASA-CRM) has been used as case of study. To summarize, the main contribution of this work lies on the development of an open-source and fully scriptable methodology, which can be easily implemented in any computational environment for aircraft design and optimization.

English

Through tropical normal idempotent matrices, we introduce isocanted alcoved polytopes, computing their f-vectors and checking the validity of the following five conjectures: Barany, unimodality, 3d, flag and cubical lower bound (CLBC). Isocanted alcoved polytopes are centrally symmetric, almost simple cubical polytopes. They are zonotopes. We show that, for each dimension, there is a unique combinatorial type. In dimension d, an isocanted alcoved polytope has 2d+1 − 2 vertices, its face lattice is the lattice of proper subsets of [d + 1] and its diameter is d + 1. They are realizations of d-elementary cubical polytopes. The f-vector of a d-dimensional isocanted alcoved polytope attains its maximum at the integer ⌊d/3⌋.

Counterexample to a Variant of a Conjecture of Ziegler Concerning a Simple Polytope and Its Dual

Problem 4.19 in Ziegler (Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)) asserts that every simple 3-dimensional polytope has the property that its dual can be constructed as the convex hull of points chosen from the facets of the original polytope. In this note we state a variant of this conjecture that requires the points to be a subset of the vertices of the original polytope, and provide a family of counterexamples for dimension $$d \ge 3$$ .

Vertex-facet assignments for polytopes

Motivated by the search for reduced polytopes, we consider the following question: For which polytopes exists a vertex-facet assignment, that is, a matching between vertices and non-incident facets, so that the matching covers either all vertices, or all facets? We provide general conditions for the existence of such an assignment. We conclude that such exist for all simple and simplicial polytopes, as well as all polytopes of dimension dle 6. We construct counterexample in all dimensions dge 7.

Almost simplicial polytopes: the lower and upper bound theorems

this is an extended abstract of the full version. We study n-vertex d-dimensional polytopes with at most one nonsimplex facet with, say, d + s vertices, called almost simplicial polytopes. We provide tight lower and upper bounds for the face numbers of these polytopes as functions of d, n and s, thus generalizing the classical Lower Bound Theorem by Barnette and Upper Bound Theorem by McMullen, which treat the case s = 0. We characterize the minimizers and provide examples of maximizers, for any d.

Toric symplectic stacks

We give an intrinsic definition of compact toric symplectic stacks, and show that they are classified by simple convex polytopes equipped with some additional combinatorial data. This generalizes Delzant's classification of compact toric symplectic manifolds. As an application, we show that any compact toric symplectic stack can be deformed to an ineffective toric orbifold.

Graph invariants and Betti numbers of real toric manifolds

For a graph $G$, the graph cubeahedron $\square_G$ and the graph associahedron $\triangle_G$ are simple convex polytopes which admit (real) toric manifolds. In this paper, we introduce a graph invariant, called the $b$-number, and show that the $b$-numbers compute the Betti numbers of the real toric manifold $X^\mathbb{R}(\square_G)$ corresponding to $\square_G$. The $b$-number is a counterpart of the notion of $a$-number, introduced by S. Choi and the second named author, which computes the Betti numbers of the real toric manifold $X^\mathbb{R}(\triangle_G)$ corresponding to $\triangle_G$. We also study various relationships between $a$-numbers and $b$-numbers from the viewpoint of toric topology. Interestingly, for a forest $G$ and its line graph $L(G)$, the real toric manifolds $X^\mathbb{R}(\triangle_G)$ and $X^\mathbb{R}(\square_{L(G)})$ have the same Betti numbers.

The topology of moment-angle manifolds—on a conjecture of S. Gitler and S. López de Medrano

In this paper, we study the topology of moment-angle manifolds and prove a conjecture of S. Gitler and S. Lopez de Medrano concerned with the behavior of the moment-angle manifold under the surgery ‘cutting off a vertex’ on a simple polytope. Let P be a simple polytope of dimension n with m facets and Pv be a polytope obtained from P by cutting off one vertex v. Let Z = Z(P) and Zv = Z(Pv) be the corresponding moment-angle manifolds. S. Gitler and S. Lopez de Medrano conjectured that: Zv is diffeomorphic to $$\partial \left[ {\left( {Z - {D^{n + m}}} \right) \times {D^2}} \right]\# \,\# _{j = 1}^{m - n}\left( {_j^{m - n}} \right)\left( {{S^{j + 2}} \times {S^{m + n - j - 1}}} \right)$$ , and they proved the conjecture in the case m < 3n. In this paper we prove the conjecture in the general case.

Three-dimensional normal pseudomanifolds with relatively few edges

Let Δ be a d-dimensional normal pseudomanifold, d≥3. A relative lower bound for the number of edges in Δ is that g2 of Δ is at least g2 of the link of any vertex. When this inequality is sharp Δ has relatively minimalg2. For example, whenever the one-skeleton of Δ equals the one-skeleton of the star of a vertex, then Δ has relatively minimal g2. Subdividing a facet in such an example also gives a complex with relatively minimal g2. We prove that in dimension three these are the only examples. As an application we determine the combinatorial and topological type of 3-dimensional Δ with relatively minimal g2 whenever Δ has two or fewer singularities. The topological type of any such complex is a pseudocompression body, a pseudomanifold version of a compression body.Complete combinatorial descriptions of Δ with g2(Δ)≤2 are due to Kalai [12](g2=0), Nevo and Novinsky [13](g2=1) and Zheng [20](g2=2). In all three cases Δ is the boundary of a simplicial polytope. Zheng observed that for all d≥0 there are triangulations of Sd⁎RP2 with g2=3. She asked if this is the only nonspherical topology possible for g2(Δ)=3. As another application of relatively minimal g2 we give an affirmative answer when Δ is 3-dimensional.

Weighted Lattice Point Sums in Lattice Polytopes, Unifying Dehn–Sommerville and Ehrhart–Macdonald

Let V be a real vector space of dimension n and let $$M\subset V$$ be a lattice. Let $$P\subset V$$ be an n-dimensional polytope with vertices in M, and let $$\varphi :V\rightarrow {\mathbb {C}}$$ be a homogeneous polynomial function of degree d. For $$q\in {\mathbb {Z}}_{>0}$$ and any face F of P, let $$D_{\varphi ,F} (q)$$ be the sum of $$\varphi $$ over the lattice points in the dilate qF. We define a generating function $$G_{\varphi }(q,y) \in {\mathbb {Q}}[q] [y]$$ packaging together the various $$D_{\varphi ,F} (q)$$ , and show that it satisfies a functional equation that simultaneously generalizes Ehrhart–Macdonald reciprocity and the Dehn–Sommerville relations. When P is a simple lattice polytope (i.e., each vertex meets n edges), we show how $$G_{\varphi }$$ can be computed using an analogue of Brion–Vergne’s Euler–Maclaurin summation formula.

Binomial Eulerian polynomials for colored permutations

Binomial Eulerian polynomials first appeared in work of Postnikov, Reiner and Williams on the face enumeration of generalized permutohedra. They are γ-positive (in particular, palindromic and unimodal) polynomials which can be interpreted as h-polynomials of certain flag simplicial polytopes and which admit interesting Schur γ-positive symmetric function generalizations. This paper introduces analogues of these polynomials for r-colored permutations with similar properties and uncovers some new instances of equivariant γ-positivity in geometric combinatorics.

Circuit walks in integral polyhedra

Circuits play a fundamental role in the theory of linear programming due to their intimate connection to algorithms of combinatorial optimization and the efficiency of the simplex method. We are interested in better understanding the properties of circuit walks in integral polyhedra. In this paper, we introduce a hierarchy for integral polyhedra based on different types of behavior exhibited by their circuit walks. Many problems in combinatorial optimization fall into the most interesting categories of this hierarchy — steps of circuit walks only stop at integer points, at vertices, or follow actual edges. We classify several classical families of polyhedra within the hierarchy, including 0/1-polytopes, polyhedra defined by totally unimodular matrices, and more specifically matroid polytopes, transportation polytopes, and partition polytopes. Finally, we prove three characterizations of the simple polytopes that appear in the bottom level of the hierarchy where all circuit walks are edge walks, showing that such polytopes constitute a generalization of simplices and parallelotopes.

Moment varieties of measures on polytopes

The uniform probability measure on a convex polytope induces piecewise polynomialdensities on its projections. For a fixed combinatorial type of simplicial polytopes, themoments of these measures a ...

Massey products, toric topology and combinatorics of polytopes

In this paper we introduce a direct family of simple polytopes such that for any there are non-trivial strictly defined Massey products of order in the cohomology rings of their moment-angle manifolds . We prove that the direct sequence of manifolds has the following properties: every manifold is a retract of , and one has inverse sequences in cohomology (over and , where as ) of the Massey products constructed. As an application we get that there are non-trivial differentials , for arbitrarily large as , in the Eilenberg–Moore spectral sequence connecting the rings and with coefficients in a field, where .