J. Lawrence, I. P. Silva
We describe a bijection between oriented cubes and adjoints of cross-polytopes. This correspondence is used to prove that the real affine cube is, up to reorientation in the same class, the unique oriented cube that is realizable. Moreover, its underlying matroid is, up to isomorphism, the unique combinatorial cube that is realizable over the reals.
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Monotone path polytopes arise as a special case of the construction of fiber polytopes, introduced by Billera and Sturmfels. A simple example is provided by the permutahedron, which is a monotone path polytope of the standard unit cube. The permutahedron is the zonotope polar to the braid arrangement. We show how the zonotopes polar to the cones of certain deformations of the braid arrangement can be realized as monotone path polytopes. The construction is an extension of that of the permutahedron and yields interesting connections between enumerative combinatorics of hyperplane arrangements a…
Geometry of Adjoint Hypersurfaces for Polytopes
本文解决Kohn和Ranestad提出的问题，证明任意凸多面体的伴随多项式由其在剩余排列（或其零维子集）上以正确阶数消失唯一确定（相差缩放），并将原定理推广到非简单多面体。
This paper continues investigation of the class of flag simple polytopes called 2-truncated cubes. It is an extended version of the short note Volodin (2012). A 2-truncated cube is a polytope obtained from a cube by sequence of truncations of codimension 2 faces. Constructed uniquely defined function which maps any 2-truncated cube to a flag simplicial complex with $f$-vector equal to $γ$-vector of the polytope. As a corollary we obtain that $γ$-vectors of 2-truncated cubes satisfy Frankl-Furedi-Kalai inequalities.
Smooth Combinatorial Cubes are IDP
本文针对Tadao Oda提出的所有光滑多面体都具有整数分解性质的猜想，证明了任意维度的光滑组合立方体这一多面体子类满足该性质。
We consider the realization space of the $d$-dimensional cube, and show that any two realizations are connected by a finite sequence of projective transformations and normal transformations. We use this fact to define an analog of the connected sum construction for cubical $d$-polytopes, and apply this construction to certain cubical $d$-polytopes to conclude that the rays spanned by $f$-vectors of cubical $d$-polytopes are dense in Adin's cone. The connectivity result on cubes extends to any product of simplices, and further, it shows the respective realization spaces are contractible.
Reconstruction of slices of cubes
本文证明立方体切片可由其图唯一确定，并构造反例说明该性质不适用于一般简单多胞体。
The main purpose of this paper is to popularize Danzer's power complex construction and establish some new results about covering maps between two power complexes. Power complexes are cube-like combinatorial structures that share many structural properties with higher-dimensional cubes and cubical tessellations on manifolds. Power complexes that are also abstract polytopes have repeatedly appeared somewhat unexpectedly in various contexts, although often under a different name. However, the non-polytope case is largely unexplored.
The linkedness of cubical polytopes: beyond the cube
本文研究立方体多胞形图的连通性，证明每个立方体 d-多胞形（d≠3）是 ⌊(d+1)/2⌋-连通的，且是强 ⌊d/2⌋-连通的，结果最优。
The linkedness of cubical polytopes: The cube
本文证明了立方体多胞体的图具有最大可能的连通性，并引入强连通性概念，回答了Wotzlaw的问题。
In the early 1990's, Billera and Sturmfels introduced the monotone path polytope (MPP), a special case of the general theory of fiber polytopes that associates a polytope to a pair $(P,φ)$ of a polytope $P$ and linear functional $φ$. In that same paper, they showed that MPPs of simplices and hyper-cubes are combinatorial cubes and permutahedra respectively. Their work has lead to many developments in combinatorics. Here we investigate the monotone paths for generic orientations of cross-polytopes. We show the face lattice of its MPP is isomorphic to the lattice of intervals in the sign poset f…
Neighborly cubical polytopes exist: for any $n\ge d\ge 2r+2$, there is a cubical convex d-polytope $C^n_d$ whose $r$-skeleton is combinatorially equivalent to that of the $n$-dimensional cube. This solves a problem of Babson, Billera & Chan. Kalai conjectured that the boundary $\partial C^n_d$ of a neighborly cubical polytope $C^n_d$ maximizes the $f$-vector among all cubical $(d-1)$-spheres with $2^n$ vertices. While we show that this is true for polytopal spheres for $n\le d+1$, we also give a counter-example for $d=4$ and $n=6$. Further, the existence of neighborly cubical polytopes shows t…
The hyperdeterminant of format 2 x 2 x 2 x 2 is a polynomial of degree 24 in 16 unknowns which has 2894276 terms. We compute the Newton polytope of this polynomial and the secondary polytope of the 4-cube. The 87959448 regular triangulations of the 4-cube are classified into 25448 D-equivalence classes, one for each vertex of the Newton polytope. The 4-cube has 80876 coarsest regular subdivisions, one for each facet of the secondary polytope, but only 268 of them come from the hyperdeterminant.
Strongly dominant weight polytopes are cubes
本文证明任意秩为 r 的根系中与强支配权 λ 对应的支配权多胞体 P^λ 均与 r 维立方体组合等价，并借此为经典型 Peterson 簇的 Betti 数公式提供了新证明。
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.
A bidirectional ballot sequence (BBS) is a finite binary sequence with the property that every prefix and suffix contains strictly more ones than zeros. BBS's were introduced by Zhao, and independently by Bosquet-M{é}lou and Ponty as $(1,1)$-culminating paths. Both sets of authors noted the difficulty in counting these objects, and to date research on bidirectional ballot sequences has been concerned with asymptotics. We introduce a continuous analogue of bidirectional ballot sequences which we call bidirectional gerrymanders, and show that the set of bidirectional gerrymanders form a convex p…
We study the vertices of the polytopes of all affine maps (a.k.a. hom-polytopes) between higher dimensional simplices, cubes, and crosspolytopes. Systematic study of general hom-polytopes was initiated in [3]. The study of such vertices is the classical aspect of a conjectural homological theory of convex polytopes. One quickly runs into open problems even for simple source and target polytopes. The vertices of Hom(simplex_m,-) and Hom(-,cube_n) are easily understood. In this work we describe the vertex sets of Hom(box_m,simplex_n), Hom(diamond_m,simplex_n), and Hom(diamond_m,diamond_n). The e…
We show how to construct homology bases for certain CW complexes in terms of discrete Morse theory and cellular homology. We apply this technique to study certain subcomplexes of the half cube polytope studied in previous works. This involves constructing explicit complete acyclic Morse matchings on the face lattice of the half cube; this procedure may be of independent interest for other highly symmetric polytopes.
Rubik's Abstract Polytopes
本文将魔方及其构型群推广至任意抽象正多面体，完整刻画任意维魔方单纯形的相关群，并概述魔方超立方体的类似结论。
Adjoints of Polytopes: Determinantal Representations and Smoothness
本文研究了多面体伴随超曲面的行列式表示，证明了所有多边形的伴随多项式可表示为三对角矩阵的行列式，并探讨了伴随超曲面的光滑性与高维表示问题。
We show that every non-degenerate regular polytope can be used to construct a thin, residually-connected, chamber-transitive incidence geometry, i.e. a regular hypertope, with a tail-triangle Coxeter diagram. We discuss several interesting examples derived when this construction is applied to generalised cubes. In particular, we produce an example of a rank $5$ finite locally spherical proper hypertope of hyperbolic type.
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.
We define an analogue of the cube and an analogue of the 5-wedge in higher dimensions, each with $2d+2$ vertices and $d^2+2d-3$ edges. We show that these two are the only minimisers of the number of edges, amongst d-polytopes with $2d+2$ vertices, for all $d$ except 4, 5 and 7. We also show that there are four sporadic minimisers in these low dimensions. We announce a partial solution to the corresponding problem for polytopes with $2d + 3$ vertices.
We show that there is a unique hypersurface of minimal degree passing through the non-faces of a polytope which is defined by a simple hyperplane arrangement. This generalizes the construction of the adjoint curve of a polygon by Wachspress in 1975. The defining polynomial of our adjoint hypersurface is the adjoint polynomial introduced by Warren in 1996. This is a key ingredient for the definition of Wachspress coordinates, which are barycentric coordinates on an arbitrary convex polytope. The adjoint polynomial also appears both in algebraic statistics, when studying the moments of uniform p…
For two elements $v$ and $w$ of the symmetric group $\mathfrak{S}_n$ with $v\leq w$ in Bruhat order, the Bruhat interval polytope $Q_{v,w}$ is the convex hull of the points $(z(1),\ldots,z(n))\in \mathbb{R}^n$ with $v\leq z\leq w$. It is known that the Bruhat interval polytope $Q_{v,w}$ is the moment map image of the Richardson variety $X^{v^{-1}}_{w^{-1}}$. We say that $Q_{v,w}$ is \emph{toric} if the corresponding Richardson variety $X_{w^{-1}}^{v^{-1}}$ is a toric variety. We show that when $Q_{v,w}$ is toric, its combinatorial type is determined by the poset structure of the Bruhat interva…
Positive Polytopes with Few Facets in the Grassmannian
本文研究 Gr(2,4) 与少面简单多面体相交所得正几何的伴随超曲面，证明五面情形唯一，并给出六面情形伴随族维数上界及非唯一仍为正几何的例子。
Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both explicitly and using Keevash's proof of existence of designs.
Canonical forms of polytopes from adjoints
本文证明尖点多面体锥射影化的正几何规范形式中，分子由对偶锥的伴随多项式唯一给出，从而解释了其消去多面体外部多余极点的作用。
The canonical form, scissors congruence and adjoint degrees of polytopes
本文在多胞体剪刀同余背景下研究典范形式，定义伴随多项式的次数下降不变量，证明其运算性质，并利用约化典范形式刻画带状多胞体，给出分解公式及推广赋值。
Originally introduced by Fine and Reid in the study of plurigenera of toric hypersurfaces, the Fine interior of a lattice polytope got recently into the focus of research. It is has been used for constructing canonical models in the sense of Mori Theory [arXiv:2008.05814]. Based on the Fine interior, we propose here a modification of the original adjoint polytopes as defined in [arXiv:1105.2415], by defining the Fine adjoint polytope $P^{F(s)}$ of $P$ as consisting of the points in $P$ that have lattice distance at least $s$ to all valid inequalities for $P$. We obtain a Fine Polyhedral Adjunc…
We prove that the intersection of a Hirsch polytope and a cube may be a non-Hirsch polytope.
Octopuses in the Boolean cube: families with pairwise small intersections, part II
本文研究布尔立方体中多个集族两两交集大小受限时集族规模乘积的极值问题，在前期渐近结果基础上，进一步给出了极值集族的强结构性刻画。
An $S$-hypersimplex for $S \subseteq \{0,1, \dots,d\}$ is the convex hull of all $0/1$-vectors of length $d$ with coordinate sum in $S$. These polytopes generalize the classical hypersimplices as well as cubes, crosspolytopes, and halfcubes. In this paper we study faces and dissections of $S$-hypersimplices. Moreover, we show that monotone path polytopes of $S$-hypersimplices yield all types of multipermutahedra. In analogy to cubes, we also show that the number of simplices in a pulling triangulation of a halfcube is independent of the pulling order.
A Counterexample to Ziegler's Cross-Polytope Conjecture for Simplicial 0/1-Polytopes
通过构造一个7维单纯0/1-多胞体（14个顶点）并证明其非中心对称，否定了Ziegler关于顶点数达到上界时必为中心对称交叉多胞体的猜想。
As a discrete analog to Minkowski's theorem on convex bodies, Wills conjectured that the Ehrhart coefficients of a centrally symmetric lattice polytope with exactly one interior lattice point are maximized by those of the cube of side length two. We discuss several counterexamples to this conjecture and, on the positive side, we identify a family of lattice polytopes that fulfill the claimed inequalities. This family is related to the recently introduced class of $l$-reflexive polytopes.
A quadratic upper bound on the Chvátal rank of polytopes in the 0/1-cube
本文证明0/1立方体中多面体的Chvátal秩至多为二次阶，改进先前上界，并借助多尺度Dirichlet逼近定理确立其紧的$\Theta(n^2)$复杂度。
We study the polytopes of affine maps between two polytopes -- the hom-polytopes. The hom-polytope functor has a left adjoint -- tensor product polytopes. The analogy with the category of vector spaces is limited, as we illustrate by a series of explicit examples exhibiting various extremal properties. The main challenge for hom-polytopes is to determine their vertices. A polytopal analogue of the rank-nullity theorem amounts to understanding how the vertex maps behave relative to their surjective and injective factors. This leads to interesting classes of surjective maps. In the last two sect…
Kissing polytopes in dimension 3
该研究确定了三维立方体中两个不相交格多胞体的最小距离公式，通过建模为格点最小化问题并利用符号计算求解多项式根来证明。
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…
The collection $\mathcal{M}_n$ of all metric spaces on $n$ points whose diameter is at most $2$ can naturally be viewed as a compact convex subset of $\mathbb{R}^{\binom{n}{2}}$, known as the metric polytope. In this paper, we study the metric polytope for large $n$ and show that it is close to the cube $[1,2]^{\binom{n}{2}} \subseteq \mathcal{M}_n$ in the following two senses. First, the volume of the polytope is not much larger than that of the cube, with the following quantitative estimates: \[ \left(\tfrac{1}{6}+o(1)\right)n^{3/2} \le \log \mathrm{Vol}(\mathcal{M}_n)\le O(n^{3/2}). \] Seco…
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