The cyclic polytope $C(n,d)$ is the convex hull of any $n$ points on the moment curve ${(t,t^2,...,t^d):t \in \reals}$ in $\reals^d$. For $d' >d$, we consider the fiber polytope (in the sense of Billera and Sturmfels) associated to the natural projection of cyclic polytopes $π: C(n,d') \to C(n,d)$ which "forgets" the last $d'-d$ coordinates. It is known that this fiber polytope has face lattice indexed by the coherent polytopal subdivisions of $C(n,d)$ which are induced by the map $π$. Our main result characterizes the triples $(n,d,d')$ for which the fiber polytope is canonical in either of the following two senses:
- all polytopal subdivisions induced by $π$ are coherent,
- the structure of the fiber polytope does not depend upon the choice of points on the moment curve.
We also discuss a new instance with a positive answer to the Generalized Baues Problem, namely that of a projection $π:P\to Q$ where $Q$ has only regular subdivisions and $P$ has two more vertices than its dimension.
Ordinary polytopes are known as a non-simplicial generalization of the cyclic polytopes. The face vectors of ordinary polytopes are shown to be log-concave.
Cyclic polytopes are generally known for being involved in the Upper Bound Theorem, but they have another extremal property which is less well known. Namely, the special shape of their f-vectors makes them applicable to certain constructions to present non-unimodal convex polytopes. Nevertheless, the f-vectors of cyclic polytopes themselves are unimodal.
Cyclic polytopes are characterized as simplicial polytopes satisfying Gale's evenness condition (a combinatorial condition on facets relative to a fixed ordering of the vertices). Periodically-cyclic polytopes are polytopes for which certain subpolytopes are cyclic. Bisztriczky discovered a class of periodically-cyclic polytopes that also satisfy Gale's evenness condition. The faces of these polytopes are braxtopes, a certain class of nonsimplicial polytopes studied by the authors. In this paper we prove that the periodically-cyclic Gale polytopes of Bisztriczky are exactly the polytopes that …
The Ehrhart polynomial of an integral convex polytope counts the number of lattice points in dilates of the polytope. In math.CO/0402148, the authors conjectured that for any cyclic polytope with integral parameters, the Ehrhart polynomial of it is equal to its volume plus the Ehrhart polynomial of its lower envelope and proved the case when the dimension d = 2. In our article, we prove the conjecture for any dimension.
Let $d$ and $n$ be positive integers with $n \geq d + 1$ and $τ_{1}, ..., τ_{n}$ integers with $τ_{1} < ... < τ_{n}$. Let $C_{d}(τ_{1}, ..., τ_{n}) \subset \RR^{d}$ denote the cyclic polytope of dimension $d$ with $n$ vertices $(τ_{1},τ_{1}^{2},...,τ_{1}^{d}), ..., (τ_{n},τ_{n}^{2},...,τ_{n}^{d})$. We are interested in finding the smallest integer $γ_{d}$ such that if $τ_{i+1} - τ_{i} \geq γ_{d}$ for $1 \leq i < n$, then $C_{d}(τ_{1}, ..., τ_{n})$ is normal. One of the known results is $γ_{d} \leq d (d + 1)$. In the present paper a new inequality $γ_{d} \leq d^{2} - 1$ is proved. Moreover, it …
There are two related poset structures, the higher Stasheff-Tamari orders, on the set of all triangulations of the cyclic $d$ polytope with $n$ vertices. In this paper it is shown that both of them have the homotopy type of a sphere of dimension $n-d-3$. Moreover, we resolve positively a new special case of the \emph{Generalized Baues Problem}: The Baues poset of all polytopal decompositions of a cyclic polytope of dimension $d \leq 3$ has the homotopy type of a sphere of dimension $n-d-2$.
We introduce several classes of polytopes contained in $[0,1]^n$ and cut out by inequalities involving sums of consecutive coordinates. We show that the normalized volumes of these polytopes enumerate circular extensions of certain partial cyclic orders. Among other things this gives a new point of view on a question popularized by Stanley. We also provide a combinatorial interpretation of the Ehrhart $h^*$-polynomials of some of these polytopes in terms of descents of total cyclic orders. The Euler numbers, the Eulerian numbers and the Narayana numbers appear as special cases.
We study polytopes defined by inequalities of the form $\sum_{i\in I} z_{i}\leq 1$ for $I\subseteq [d]$ and nonnegative $z_i$ where the inequalities can be reordered into a matrix inequality involving a column-convex $\{0,1\}$-matrix. These generalize polytopes studied by Stanley, and the consecutive coordinate polytopes of Ayyer, Josuat-Vergès, and Ramassamy. We prove an integral equivalence between these polytopes and flow polytopes of directed acyclic graphs $G$ with a Hamiltonian path, which we call spinal graphs. We show that the volume of these flow polytopes is the number of extensions …
In the present paper, we consider the problem when the toric ring arising from an integral cyclic polytope is Cohen-Macaulay by discussing Serre's condition and we give a complete characterization when that is Gorenstein. Moreover, we study the normality of the other semigroup ring arising from an integral cyclic polytope but generated only with its vertices.
We report on enumerating the triangulations of cyclic polytopes with the new software mptopcom. This is relevant for its connection with higher Stasheff-Tamari orders, which occur in category theory and algebraic combinatorics.
We investigate the combinatorics and geometry of permutation polytopes associated to cyclic permutation groups, i.e., the convex hulls of cyclic groups of permutation matrices. We give formulas for their dimension and vertex degree. In the situation that the generator of the group consists of at most two orbits, we can give a complete combinatorial description of the associated permutation polytope. In the case of three orbits the facet structure is already quite complex. For a large class of examples we show that there exist exponentially many facets.
The simplex algorithm using the random edge pivot-rule on any realization of a dual cyclic 4-polytope with n facets does not take more than O(n) pivot-steps. This even holds for general abstract objective functions (AOF) / acyclic unique sink orientations (AUSO). The methods can be used to show analogous results for products of two polygons. In contrast, we show that the random facet pivot-rule is slow on dual cyclic 4-polytopes, i.e. there are AUSOs on which random facet takes at least Ω(n^2) steps.
We prove a bijection between the triangulations of the 3-dimensional cyclic polytope C(n+2, 3) and persistent graphs with n vertices. We show that under this bijection the Stasheff-Tamari orders on triangulations naturally translate to subgraph inclusion between persistent graphs. Moreover, we describe a connection to the second higher Bruhat order B(n, 2). We additionally give an algorithm to efficiently enumerate all persistent graphs on n vertices and thus all triangulations of C(n+2, 3).
We define a centrally symmetric analogue of the cyclic polytope and study its facial structure. We conjecture that our polytopes provide asymptotically the largest number of faces in all dimensions among all centrally symmetric polytopes with n vertices of a given even dimension d=2k when d is fixed and n grows. For a fixed even dimension d=2k and an integer 0< j <k we prove that the maximum possible number of j-dimensional faces of a centrally symmetric d-dimensional polytope with n vertices is at least (c_j(d)+o(1)) {n \choose j+1} for some c_j(d)>0 and at most (1-2^{-d}+o(1)){n \choose j+1}…
Neighborly cubical polytopes are known as the cubical analogues of the cyclic polytopes. Using the short cubical $h$-vectors of cubical polytopes (introduced by Adin), we derive an explicit formula for the face numbers of the neighborly cubical polytopes. These face numbers form a unimodal sequence.
We study the structure of Stanley-Reisner rings associated to cyclic polytopes, using ideas from unprojection theory. Consider the boundary simplicial complex Delta(d,m) of the d-dimensional cyclic polytope with m vertices. We show how to express the Stanley-Reisner ring of Delta(d,m+1) in terms of the Stanley-Reisner rings of Delta(d,m) and Delta(d-2,m-1). As an application, we use the Kustin-Miller complex construction to identify the minimal graded free resolutions of these rings. In particular, we recover results of Schenzel, Terai and Hibi about their graded Betti numbers.
The software TrIm offers implementations of tropical implicitization and tropical elimination, as developed by Tevelev and the authors. Given a polynomial map with generic coefficients, TrIm computes the tropical variety of the image. When the image is a hypersurface, the output is the Newton polytope of the defining polynomial. TrIm can thus be used to compute mixed fiber polytopes, including secondary polytopes.
This paper focuses on determining the volumes of permutation polytopes associated to cyclic groups, dihedral groups, groups of automorphisms of tree graphs, and Frobenius groups. We do this through the use of triangulations and the calculation of Ehrhart polynomials. We also present results on the theta body hierarchy of various permutation polytopes.
Any totally positive $(k+m)\times n$ matrix induces a map $π_+$ from the positive Grassmannian ${\rm Gr}_+(k,n)$ to the Grassmannian ${\rm Gr}(k,k+m)$, whose image is the amplituhedron $\mathcal{A}_{n,k,m}$ and is endowed with a top-degree form called the canonical form ${\bfΩ}(\mathcal{A}_{n,k,m})$. This construction was introduced by Arkani-Hamed and Trnka, where they showed that ${\bfΩ}(\mathcal{A}_{n,k,4})$ encodes scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills theory. Moreover, the computation of ${\bfΩ}(\mathcal{A}_{n,k,m})$ is reduced to finding the triangulations of $\mathca…
A seminal result of E. Ehrhart states that the number of integer lattice points in the dilation of a rational polytope by a positive integer $k$ is a quasi-polynomial function of $k$ --- that is, a "polynomial" in which the coefficients are themselves periodic functions of $k$. Using a result of F. Liu on the Ehrhart polynomials of cyclic polytopes, we construct not-necessarily-convex rational polytopes of arbitrary dimension in which the periods of the coefficient functions appearing in the Ehrhart quasi-polynomial take on arbitrary values.
A cellular string of a polytope is a sequence of faces stacked on top of each other in a given direction. The poset of cellular strings, ordered by refinement, is known to be homotopy equivalent to a sphere. The subposet of coherent cellular strings is the face lattice of the fiber polytope, hence is homeomorphic to a sphere. In some special cases, every cellular string is coherent. Such polytopes are said to be all-coherent. We give a complete classification of zonotopes with the all-coherence property in terms of their oriented matroid structure. Although the face lattice of the fiber polyto…
This note studies the geometric structure of monotone moment polytopes (the duals of smooth Fano polytopes) using probes. The latter are line segments that enter the polytope at an interior point of a facet and whose direction is integrally transverse to this facet. A point inside the polytope is displaceable by a probe if it lies less than half way along it. Using a construction due to Fukaya-Oh-Ohta-Ono, we show that every rational polytope has a central point that is not displaceable by probes. In the monotone (or more generally, the reflexive) case, this central point is its unique interio…
Graphs on integer points of polytopes whose edges come from a set of allowed differences are studied. It is shown that any simple graph can be embedded in that way. The minimal dimension of such a representation is the fiber dimension of the given graph. The fiber dimension is determined for various classes of graphs and an upper bound in terms of the chromatic number is stated.
If we fix the angles at the vertices of a convex planar $n$-gon, the lengths of its edges must satisfy two linear constraints in order for it to close up. If we also require unit perimeter, our vectors of $n$ edge lengths form a convex polytope of dimension $n-3$, each facet of which consists of those $n$-gons in which the length of a particular edge has fallen to zero. Bavard and Ghys require unit area instead, which gives them a hyperbolic polytope. Those two polytopes are combinatorially equivalent, so either is fine for our purposes. Such a fixed-angles polytope is combinatorially richer w…
There is a simple formula for the Ehrhart polynomial of a cyclic polytope. The purpose of this paper is to show that the same formula holds for a more general class of polytopes, lattice-face polytopes. We develop a way of decomposing any d-dimensional simplex in general position into d! signed sets, each of which corresponds to a permutation in the symmetric group, and reduce the problem of counting lattice points in a polytope in general position to that of counting lattice points in these special signed sets. Applying this decomposition to a lattice-face simplex, we obtain signed sets with …
For a lattice path $ν$ from the origin to a point $(a,b)$ using steps $E=(1,0)$ and $N=(0,1)$, we construct an associated flow polytope $\mathcal{F}_{\hat{G}_B(ν)}$ arising from an acyclic graph where bidirectional edges are permitted. We show that the flow polytope $\mathcal{F}_{\hat{G}_B(ν)}$ admits a subdivision dual to a $w$-simplex, where $w$ is the number of valleys in the path $\barν = EνN$. Refinements of this subdivision can be obtained by reductions of a polynomial $P_ν$ in a generalization of Mészáros' subdivision algebra for acyclic root polytopes where negative roots are allowed. …
Higher Auslander algebras were introduced by Iyama generalizing classical concepts from representation theory of finite dimensional algebras. Recently these higher analogues of classical representation theory have been increasingly studied. Cyclic polytopes are classical objects of study in convex geometry. In particular, their triangulations have been studied with a view towards generalizing the rich combinatorial structure of triangulations of polygons. In this paper, we demonstrate a connection between these two seemingly unrelated subjects. We study triangulations of even-dimensional cycli…
Bott-Samelson varieties are a twisted product of $\mathbb{C}\mathbb{P}^1$'s with a map into $G/B$. These varieties are mostly studied in the case in which the map into $G/B$ is birational to the image; however in this paper we study a fiber of this map when it is not birational. We will see that in some cases the general fiber, which we christen a brick manifold, is a toric variety. In order to do so we use the moment map of a Bott-Samelson variety to translate this problem into one in terms of the "subword complexes" of Knutson and Miller. Pilaud and Stump realized certain subword complexes a…
We construct, for any positive integer n, a family of n congruent convex polyhedra in R^3, such that every pair intersects in a common facet. Previously, the largest such family contained only eight polytopes. Our polyhedra are Voronoi regions of evenly distributed points on the helix (t, cos t, sin t). With a simple modification, we can ensure that each polyhedron in the family has a point, a line, and a plane of symmetry. We also generalize our construction to higher dimensions and introduce a new family of cyclic polytopes.
Let $π:{\mathbb R}^n \to {\mathbb R}^d$ be any linear projection, let $A$ be the image of the standard basis. Motivated by Postnikov's study of postitive Grassmannians via plabic graphs and Galashin's connection of plabic graphs to slices of zonotopal tilings of 3-dimensional cyclic zonotopes, we study the poset of subdivisions induced by the restriction of $π$ to the $k$-th hypersimplex, for $k=1,\dots,n-1$. We show that: - For arbitrary $A$ and for $k\le d+1$, the corresponding fiber polytope $\mathcal F^{(k)}(A)$ is normally isomorphic to the Minkowski sum of the secondary polytopes of all …
In this paper we prove that points in the space $X(k,n)$ of configurations of $n$ points in $\mathbb{CP}^{k-1}$ which are fixed under a certain cyclic action are the solutions to the generalized scattering equations on planar kinematics (PK). In the first part, we give a constructive upper bound: we show that these solutions inject into certain aperiodic k-element subsets of $\{1,\ldots, n\}$, and consequently that their number is bounded above by the number of Lyndon words with k one's and n-k zeros. The proof uses a somewhat surprising connection between the superpotential of the mirror of $…
In this paper we study various scribability problems for polytopes. We begin with the classical $k$-scribability problem proposed by Steiner and generalized by Schulte, which asks about the existence of $d$-polytopes that cannot be realized with all $k$-faces tangent to a sphere. We answer this problem for stacked and cyclic polytopes for all values of $d$ and $k$. We then continue with the weak scribability problem proposed by Grünbaum and Shephard, for which we complete the work of Schulte by presenting non weakly circumscribable $3$-polytopes. Finally, we propose new $(i,j)$-scribability pr…
We investigate the combinatorics of quivers that arise from triangulations of even-dimensional cyclic polytopes. Work of Oppermann and Thomas pinpoints such quivers as the prototypes for higher-dimensional cluster theory. We first show that a $2d$-dimensional triangulation has no interior $(d + 1)$-simplices if and only if its quiver is a cut quiver of type $A$, in the sense of Iyama and Oppermann. This is a higher-dimensional generalisation of the fact that triangulations of polygons with no interior triangles correspond to orientations of an $A_{n}$ Dynkin diagram. An application of this fir…
Motivated by the Milnor fiber of a central arrangement, we study the cohomology of a family of cyclic covers of the complement of an arbitrary arrangement. We give an explicit proof of the polynomial periodicity of the Betti numbers of the members of this family of cyclic covers.
Sometimes, it is possible to represent a complicated polytope as a projection of a much simpler polytope. To quantify this phenomenon, the extension complexity of a polytope $P$ is defined to be the minimum number of facets of a (possibly higher-dimensional) polytope from which $P$ can be obtained as a (linear) projection. This notion is motivated by its relevance to combinatorial optimisation, and has been studied intensively for various specific polytopes associated with important optimisation problems. In this paper we study extension complexity as a parameter of general polytopes, more spe…
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…
A centrally symmetric analogue of the cyclic polytope, the bicyclic polytope, was defined in [BN08]. The bicyclic polytope is defined by the convex hull of finitely many points on the symmetric moment curve where the set of points has a symmetry about the origin. In this paper, we study the Barvinok-Novik orbitope, the convex hull of the symmetric moment curve. It was proven in [BN08] that the orbitope is locally $k$-neighborly, that is, the convex hull of any set of $k$ distinct points on an arc of length not exceeding $ϕ_k$ in $\mathbb{S}^1$ is a $(k-1)$-dimensional face of the orbitope for …
The celebrated upper bound theorem of McMullen determines the maximal number of extreme points of a polyhedron in terms of its dimension and the number of constraints which define it, showing that the maximum is attained by the polar of the cyclic polytope. We show that the same bound is valid in the tropical setting, up to a trivial modification. Then, we study the natural candidates to be the maximizing polyhedra, which are the polars of a family of cyclic polytopes equipped with a sign pattern. We construct bijections between the extreme points of these polars and lattice paths depending on…
In 1996, Edelman and Reiner defined the two higher Stasheff--Tamari orders on triangulations of cyclic polytopes and conjectured them to coincide. We open up an algebraic angle for approaching this conjecture by showing how these orders arise naturally in the representation theory of the higher Auslander algebras of type $A$, denoted $A_{n}^{d}$. For this we give new combinatorial interpretations of the orders, making them comparable. We then translate these combinatorial interpretations into the algebraic framework. We also show how triangulations of odd-dimensional cyclic polytopes arise in …
