Nicolau C. Saldanha, Carlos Tomei
We compute the spectra of the adjacency matrices of the semi-regular polytopes. A few different techniques are employed: the most sophisticated, which relates the 1-skeleton of the polytope to a Cayley graph, is based on methods akin to those of Lovász and Babai ([L], [B]). It turns out that the algebraic degree of the eigenvalues is at most 5, achieved at two 3-dimensional solids.
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An abstract polytope is \emph{flat} if every facet is incident on every vertex. In this paper, we prove that no chiral polytope has flat finite regular facets and finite regular vertex-figures. We then determine the three smallest non-flat regular polytopes in each rank, and use this to show that for $n \geq 8$, a chiral $n$-polytope has at least $48(n-2)(n-2)!$ flags.
Regular incidence complexes are combinatorial incidence structures generalizing regular convex polytopes, regular complex polytopes, various types of incidence geometries, and many other highly symmetric objects. The special case of abstract regular polytopes has been well-studied. The paper describes the combinatorial structure of a regular incidence complex in terms of a system of distinguished generating subgroups of its automorphism group or a flag-transitive subgroup. Then the groups admitting a flag-transitive action on an incidence complex are characterized as generalized string C-group…
In this paper we give several criteria for the edge polytope of a fundamental FHM-graph to possess a regular unimodular triangulation in terms of some simple data of the the graph. We further apply our criteria to several examples of graphs and show that their edge polytopes possess a regular unimodular triangulation.
Up to isomorphism and duality, there are exactly two non-degenerate abstract regular polytopes of rank greater than $n-3$, one of rank $n-1$ and one of rank $n-2$, with automorphism groups that are transitive permutation groups of degree $n\geq 7$. In this paper we extend this classification of high rank regular polytopes to include the ranks $n-3$ and $n-4$. The result is, up to a isomorphism and duality, seven abstract regular polytopes of rank $n-3$ for each $n\geq 9$, and nine abstract regular polytopes of rank $n-4$ for each $n \geq 11$. Moreover we show that if a transitive permutation g…
In this paper, we construct an infinite families of group $G$ of order $4p^m$ which can be an automorphism group of some regular polytope with type $\{p, 2p\}$, where $m \geq 3$ and $p$ is an odd prime. For $p=3$, our polytopes are the regular toroidal maps $\{3, 6\}$.
In this short note we show explicitly how to decompose a generalized permutohedron into semi-polytopes.
Every equivelar abstract polytope of type $\{p_1, \ldots, p_{n-1}\}$ has at least $2p_1 \cdots p_{n-1}$ flags. Polytopes that attain this lower bound are called tight. Here we investigate the question of under what conditions there is a tight orientably-regular polytope of type $\{p_1, \ldots, p_{n-1}\}$. We show that it is necessary and sufficient that whenever $p_i$ is odd, both $p_{i-1}$ and $p_{i+1}$ are even divisors of $2p_i$.
In this paper, we classify regular polytopes with automorphism groups of order $2^n$ and Schläfli types $\{4, 2^{n-3}\}, \{4, 2^{n-4}\}$ and $\{4, 2^{n-5}\}$ for $n \geq 10$, therefore giving a partial answer to a problem proposed by Schulte and Weiss in [Problems on polytopes, their groups, and realizations, Periodica Math. Hungarica 53(2006) 231-255].
The Ehrhart function $L_P(t)$ of a polytope $P$ is usually defined only for integer dilation arguments $t$. By allowing arbitrary real numbers as arguments we may also detect integer points entering (or leaving) the polytope in fractional dilations of $P$, thus giving more information about the polytope. Nevertheless, there are some polytopes that only gain new integer points for integer values of $t$; that is, these polytopes satisfy $L_P(t) = L_P(\lfloor t \rfloor)$. We call those polytopes semi-reflexive. In this paper, we give a characterization of these polytopes in terms of their hyperpl…
Chiral extensions of regular toroids
该研究针对抽象多面体中的手性结构问题，通过构建面为n维环面正则立方铺砌的手性多面体，证明了对任意d≥3存在无穷多手性d维多面体，丰富了手性多面体的存在性结论。
We prove that any finite, abstract n-polytope is covered by a finite, abstract regular n-polytope.
We prove that any simple polytope (and some non-simple polytopes) in $\mathbb R^3$ admits an inscribed regular octahedron.
Adjacency polytopes, a.k.a. symmetric edge polytopes, associated with undirected graphs have been defined and studied in several seemingly independent areas including number theory, discrete geometry, and dynamical systems. In particular, the authors are motivated by the tropical intersections problem derived from the Kuramoto equations. Regular subdivisions of adjacency polytopes are instrumental in solving these problems. This paper explores connections between the regular subdivisions of an adjacency polytope and the contraction of the underlying graph along an edge. We construct a special …
We present a multivariate generating function for all n x n nonnegative integral matrices with all row and column sums equal to a positive integer t, the so called semi-magic squares. As a consequence we obtain formulas for all coefficients of the Ehrhart polynomial of the polytope B_n of n x n doubly-stochastic matrices, also known as the Birkhoff polytope. In particular we derive formulas for the volumes of B_n and any of its faces.
Unlike the situation in the classical theory of convex polytopes, there is a wealth of semi-regular abstract polytopes, including interesting examples exhibiting some unexpected phenomena. We prove that even an equifacetted semi-regular abstract polytope can have an arbitrary large number of flag orbits or face orbits under its combinatorial automorphism group.
This paper introduces the notion of an unravelled abstract regular polytope, and proves that $\SL_3(q) \rtimes <t>$, where $t$ is the transpose inverse automorphism of $\SL_3(q)$, possesses such polytopes for various congruences of $q$. A large number of small examples of such polytopes are given, along with extensive details of their various properties.
The Birkhoff polytope, defined to be the convex hull of $n\times n$ permutation matrices, is a well studied polytope in the context of the Ehrhart theory. This polytope is known to have many desirable properties, such as the Gorenstein property and existence of regular, unimodular triangulations. In this paper, we study analogues of the Birkhoff polytope for finite irreducible Coxeter groups of other types. We focus on a type-$B$ Birkhoff polytope $\mathcal{BB}(n)$ arising from signed permutation matrices and prove that it and its dual polytope are reflexive, and hence Gorenstein, and also pos…
This paper exhibits two families of unravelled abstract regular polytopes in Coxeter groups of type Bn. For one family they have rank 4 while the other family has arbitrarily large rank.
A concept of generalized regular polytope is introduced in this work. The number of its (1...n-1)-dimensional elements is not necessarily integer, though all the combinatorial and metric properties meet those of regular polytopes in a classic sense. New relationships between Schlafli symbol of the regular polytope and its metric parameters have been established. Using the generalized regular polytopes concept, group and metric properties of arbitrary metric space tessellations into regular honeycombs were investigated. It has been shown that sequential tessellations of space into regular honey…
A Regular Unimodular Triangulation of the Matroid Base Polytope
本文构造了任意拟阵基多面体的首个正则幺模三角剖分，并将其推广至整广义置换多面体，解决了此前幺模覆盖存在性未知的问题。
In this paper for any dimension n we give a complete list of lattice convex polytopes in R^n that are regular with respect to the group of affine transformations preserving the lattice.
We investigate polyhedral $2k$-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex {\it $k$-Hamiltonian} if it contains the full $k$-skeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so-called {\it super-neighborly triangulations}) we focus on the case of the cross polytope and the sporadic regular 4-polytopes. By our results the existence of 1-Hamiltonian surfaces is now decided for all regular polytopes. Furthermore we investigate 2-Hamiltonian 4-manifolds in…
Wythoff's construction associates a uniform polytope to a Coxeter diagram whose vertices are decorated with crosses, which indicate the subgroup stabilizing a generic point. Champagne, Kjiri, Patera, and Sharp remarked that by associating more information to a Coxeter diagram, one can furthermore determine the types and number of faces of such a polytope. This article provides a proof of this result, and uses it to provide a classification of the regular polytopes.
We present a simple proof of the fact that the base (and independence) polytope of a rank $n$ regular matroid over $m$ elements has an extension complexity $O(mn)$.
On a generalization of symmetric edge polytopes to regular matroids
本文将对称边多胞体推广到正则拟阵，证明其幺模等价性由图形拟阵决定，并给出广义多胞体的面描述、显式正则幺模三角剖分及其极多胞体 Ehrhart 理论与对偶正则拟阵流格的联系。
Regular $3$-polytopes of order $2^np$
针对Schulte和Weiss提出的刻画阶为$2^np$的正则多胞体的问题，本文推导了阶为$2^np$的正则3-多胞体的Schläfli型分类，并给出两类型下多胞体存在的参数条件及相关一般构造。
Unfolding Overlaps of the Exceptional Regular Polytopes
该研究找到三种特殊四维多胞体的脊展开方式会产生面重叠，终止了具有全网属性的正多胞体的完整分类。
The ancillary file Corrigendum.pdf contains an explanation why our proof does not work.
For each $d\geq 3$, $n \geq 10$, and $k_1, k_2, \ldots, k_{d-1}\geq 2$ with $k_1+k_2+\ldots+k_{d-1}\leq n-1$, we construct a regular $d$-polytope whose automorphism group is of order $2^n$ and whose Schläfli type is $\{2^{k_1},2^{k_2}, \ldots, 2^{k_{d-1}}\}$.
In the classical setting, a convex polytope is said to be semiregular if its facets are regular and its symmetry group is transitive on vertices. This paper studies semiregular abstract polytopes, which have abstract regular facets, still with combinatorial automorphism group transitive on vertices. We analyze the structure of the automorphism group, focusing in particular on polytopes with two kinds of regular facets occurring in an "alternating" fashion. In particular we use group amalgamations to prove that given two compatible n-polytopes P and Q, there exists a universal abstract semiregu…
Given a chiral d-polytope K with regular facets, we describe a construction for a chiral (d + 1)-polytope P with facets isomorphic to K. Furthermore, P is finite whenever K is finite. We provide explicit examples of chiral 4-polytopes constructed in this way from chiral toroidal maps.
Equatorial Flow Triangulations of Gorenstein Flow Polytopes
本文应用Bruns和Römer的赤道三角剖分技术，给出Gorenstein流多面体未被研究的三角剖分的纯组合描述，证明其与框架法所得三角剖分不同，还找到投影自反多面体的面描述，并在强平面偏序集下生成序多面体新幺模三角剖分。
On the order of 4-dimensional regular polytope numbers
本文基于Kim猜想与Waring问题的推广，建立了将足够大整数表示为4维正多胞体数之和的渐近公式，并推广至满足特定条件的任意四次多项式。
Regular polytopes, sphere packings and Apollonian sections
本文研究了由任意维度正多胞体诱导的多胞体球填充的几何与算术性质，证明了所有整多胞体皆为晶体学的且共有11个晶体学正多胞体，引入阿波罗尼奥斯截面概念以确定相关截面填充，并计算了所有正多胞体的Möbius谱。
Abstract Regular Polytopes of Finite Irreducible Coxeter Groups
本文研究了有限不可约Coxeter群的抽象正多面体，确定了其最大秩及不同秩的正多面体存在性。
Semiregular abstract polyhedra with trivial facet stabilizer
本文构造了具有平凡面稳定化子的半正抽象多面体，展示了其可拥有无界的旗轨道数，同时仅有一个面轨道，并通过操作高阶正则和非正则多面体实现交替半正多面体的构造。
There exists just one regular polytope of rank larger than 3 whose full automorphism group is a projective general linear group PGL_2(q), for some prime-power q. This polytope is the 4-simplex and the corresponding group is PGL_2(5), which is isomorphic to S_5.
This article announces the completion of the classification of rank 4 locally projective polytopes and their quotients. There are seventeen universal locally projective polytopes (nine nondegenerate). Amongst their 441 quotients are a further four (nonuniversal) regular polytopes, and 152 nonregular but section regular polytopes. All 156 of the latter have hemidodecahedral facets or hemi-icosahedral vertex figures. It is noted that, remarkably, every rank 4 locally projective section regular polytope is finite. The article gives a survey of the literature of locally projective polytopes and th…
Consider a lattice in a real finite dimensional vector space. Here, we are interested in the lattice polytopes, that is the convex hulls of finite subsets of the lattice. Consider the group $G$ of the affine real transformations which map the lattice onto itself. Replacing the group of euclidean motions by the group $G$ one can define the notion of regular lattice polytopes. More precisely, a lattice polytope is said to be regular if the subgroup of $G$ which preserves the polytope acts transitively on the set of its complete flags. Recently, Karpenkov obtained a classification of the regular …
Over a decade ago, it was shown that every edge unfolding of the Platonic solids was without self-overlap, yielding a valid net. We consider this property for regular polytopes in arbitrary dimensions, notably the simplex, cube, and orthoplex. It was recently proven that all unfoldings of the $n$-cube yield nets. We show this is also true for the $n$-simplex and the $4$-orthoplex but demonstrate its surprising failure for any orthoplex of higher dimension.
There are two main thrusts in the theory of regular and chiral polytopes: the abstract, purely combinatorial aspect, and the geometric one of realizations. This brief survey concentrates on the latter. The dimension of a faithful realization of a finite abstract regular polytope in some euclidean space is no smaller than its rank, while that of a chiral polytope must strictly exceed the rank. There are similar restrictions on the dimensions of realizations of regular and chiral apeirotopes. From the viewpoint of realizations in a fixed dimension, the problems are now completely solved in up to…
We obtain computational hardness results for f-vectors of polytopes by exhibiting reductions of the problems DIVISOR and SEMI-PRIME TESTABILITY to problems on f-vectors of polytopes. Further, we show that the corresponding problems for f-vectors of simplicial polytopes are polytime solvable. The regime where we prove this computational difference (conditioned on standard conjectures on the density of primes and on $P\neq NP$) is when the dimension $d$ tends to infinity and the number of facets is linear in $d$.
Every finite, self-dual, regular (or chiral) 4-polytope of type {3,q,3} has a trivalent 3-transitive (or 2-transitive) medial layer graph. Here, by dropping self-duality, we obtain a construction for semisymmetric trivalent graphs (which are edge- but not vertex-transitive). In particular, the Gray graph arises as the medial layer graph of a certain universal locally toroidal regular 4-polytope.
Regular polytopes of rank $n/2$ for transitive groups of degree $n$
该研究针对次数为$n$的传递群对应的抽象正多胞体，探究秩为$n/2$（$n\neq14$）的情况，发现秩较最大值降低1时，正多胞体数量会大幅增加，补充了相关领域的研究结论。
Regular 3-polytopes of type $\{n,n\}$
本文对每个 $n\geq3$ 构造了含 $2^n n$ 个旗标、类型为 $\{n,n\}$ 的自对偶正则 3-多面体，并显式给出其半直积自同构群，解决了相关存在性问题。
Inspired by Coxeter's notion of Petrie polygon for $d$-polytopes (see \cite{Cox73}), we consider a generalization of the notion of zigzag circuits on complexes and compute the zigzag structure for several interesting families of $d$-polytopes, including semiregular, regular-faced, Wythoff Archimedean ones, Conway's 4-polytopes, half-cubes, folded cubes. Also considered are regular maps and Lins triality relations on maps.
We consider a Hamiltonian decomposition problem of partitioning a regular graph into edge-disjoint Hamiltonian cycles. A sufficient condition for vertex adjacency in the 1-skeleton of the traveling salesperson polytope can be formulated as the Hamiltonian decomposition problem in a 4-regular multigraph. We introduce a heuristic general variable neighborhood search algorithm for this problem based on finding a vertex-disjoint cycle cover of the multigraph through reduction to perfect matching and several cycle merging operations. The algorithm has a one-sided error: the answer "not adjacent" is…
Many regular triangulations and many polytopes
本文证明了对于固定的 $d>3$ 且 $n$ 趋于无穷，具有 $n$ 个顶点的不同标号组合型 $d$-多胞形至少有 $(n!)^{d-2 \pm o(1)}$ 种，约为先前最佳下界的平方，并以此类多胞形的正则三角剖分数量作为中间步骤。
Associahedra minimize $f$-vectors of secondary polytopes of planar point sets
本文推广了关于平点集三角剖分数量的下界结果，证明正则三角剖分已足够，并将结论扩展到所有正则细分，按次级多面体中面的维度分级。
Motivated by the study of polytopes formed as the convex hull of permutation matrices and alternating sign matrices, we define several new families of polytopes as convex hulls of sign matrices, which are certain {0,1,-1}-matrices in bijection with semistandard Young tableaux. We investigate various properties of these polytopes, including their inequality descriptions, vertices, facets, and face lattices, as well as connections to alternating sign matrix polytopes and transportation polytopes.
Answers to questions about medial layer graphs of self-dual regular and chiral polytopes
本文解答了自对偶4维多胞体中间层图与其覆盖Cayley图关系的未决问题，给出首批无极性的{3,q,3}型非真自对偶手性多胞体实例，证明p=3时Cayley图s-弧传递度不高于中间层图，并构造了一族{6,q,6}型正则多胞体满足Cayley图顶点稳定子更大。
Every regular polytope has the remarkable property that it inherits all symmetries of each of its facets. This property distinguishes a natural class of polytopes which are called hereditary. Regular polytopes are by definition hereditary, but the other polytopes in this class are interesting, have possible applications in modeling of structures, and have not been previously investigated. This paper establishes the basic theory of hereditary polytopes, focussing on the analysis and construction of hereditary polytopes with highly symmetric faces.
We construct infinite families of abstract regular polytopes of type $\{4,p_1,\ldots,p_{n-1}\}$ from extensions of centrally symmetric spherical abstract regular $n$-polytopes. In addition, by applying the halving operation, we obtain infinite families of both locally spherical and locally toroidal regular hypertopes of type $\left\{{p_1 \atop p_1},\ldots,p_{n-1}\right\}$.
Given a configuration $A$ of $n$ points in $\mathbb{R}^{d-1}$, we introduce the higher secondary polytopes $Σ_{A,1},\dots, Σ_{A,n-d}$, which have the property that $Σ_{A,1}$ agrees with the secondary polytope of Gelfand--Kapranov--Zelevinsky, while the Minkowski sum of these polytopes agrees with Billera--Sturmfels' fiber zonotope associated with (a lift of) $A$. In a special case when $d=3$, we refer to our polytopes as higher associahedra. They turn out to be related to the theory of total positivity, specifically, to certain combinatorial objects called plabic graphs, introduced by the seco…
The paper studies modular reduction techniques for abstract regular and chiral polytopes, with two purposes in mind: first, to survey the literature about modular reduction in polytopes; and second, to apply modular reduction, with moduli given by primes in Z[t] (with t=τthe golden ratio), to construct new regular 4-polytopes of hyperbolic types {3,5,3} and {5,3,5} with automorphism groups given by finite orthogonal groups.
We show that there are simple 4-dimensional polytopes with n vertices such that all separators of the graph have size at least $Ω(n/\log n)$. This establishes a strong form of a claim by Thurston, for which the construction and proof had been lost. We construct the polytopes by cutting off the vertices and then the edges of a particular type of neighborly cubical polytopes. The graphs of simple polytopes thus obtained are 4-regular; they contain 3-regular "cube-connected cycle graphs" as minors of spanning subgraphs.
Regular dyadic triangulations of delta-matroid polytopes
本文证明每个delta-拟阵多面体（及更一般的整B型广义置换多面体）存在正则二进三角剖分，其最大单纯形体积为2的幂，并指出B型中唯一非幺模单纯形例外，基于B型根配置的完全二进性。
In 1967, Grünmbaum conjectured that any $d$-dimensional polytope with $d+s\leq 2d$ vertices has at least \[ϕ_k(d+s,d) = {d+1 \choose k+1 }+{d \choose k+1 }-{d+1-s \choose k+1 } \] $k$-faces. This conjecture along with the characterization of equality cases was recently proved by the author. In this paper, several extensions of this result are established. Specifically, it is proved that lattices with the diamond property (for example, abstract polytopes) and $d+s\leq 2d$ atoms have at least $ϕ_k(d+s,d)$ elements of rank $k+1$. Furthermore, in the case of face lattices of strongly regular CW co…
Semi-interlaced polytopes
本文研究半交错多面体的混合体积，提出一种组合公式，并应用于Arnold单调性问题。
We present the results of an investigation into the representations of Archimedean polyhedra (those polyhedra containing only one type of vertex figure) as quotients of regular abstract polytopes. Two methods of generating these presentations are discussed, one of which may be applied in a general setting, and another which makes use of a regular polytope with the same automorphism group as the desired quotient. Representations of the 14 sporadic Archimedean polyhedra (including the pseudorhombicuboctahedron) as quotients of regular abstract polyhedra are obtained, and summarised in a table. T…
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