We describe a construction for d-polytopes generalising the well known stacking operation. The construction is applied to produce 2-simplicial and 2-simple 4-polytopes with g_2=0 on any number of n >= 13 vertices. In particular, this implies that the ray l_1, described by Bayer (1987), is fully contained in the convex hull of all flag vectors of 4-polytopes. Especially interesting examples on 9, 10 and 11 vertices are presented.
In this paper, we study flag structures of matroid base polytopes. We describe faces of matroid base polytopes in terms of matroid data, and give conditions for hyperplane splits of matroid base polytopes. Also, we show how the cd-index of a polytope can be expressed when a polytope is split by a hyperplane, and apply these to the cd-index of a matroid base polytope of a rank 2 matroid.
Bisztriczky introduced the multiplex as a generalization of the simplex. A polytope is multiplicial if all its faces are multiplexes. In this paper it is proved that the flag vectors of multiplicial polytopes depend only on their face vectors. A special class of multiplicial polytopes, also discovered by Bisztriczky, is comprised of the ordinary polytopes. These are a natural generalization of the cyclic polytopes. The flag vectors of ordinary polytopes are determined. This is used to give a surprisingly simple formula for the h-vector of the ordinary d-polytope with n+1 vertices and character…
An abstract polytope is chiral if its automorphism group has two orbits on the flags, such that adjacent flags belong to distinct orbits. There are still few examples of chiral polytopes, and few constructions that can create chiral polytopes with specified properties. In this paper, we show how to build self-dual chiral polytopes using the mixing construction for polytopes.
In the focus of this paper is the operation of edge contraction. One can show that simplicial 3-polytope is flag iff contraction of any its edge gives simplicial 3-polytope. Our main result states that any flag simplicial 3-polytope can be reduced to octahedron by sequence of edge contractions. Using this operation we introduce a partial order on the set of flag simplicial 3-polytopes and study Hasse graph of corresponding poset. We estimate input and output degrees of vertices of this Hasse graph.
An abstract polytope of rank n is said to be chiral if its automorphism group has two orbits on the flags, such that adjacent flags belong to distinct orbits. Examples of chiral polytopes have been difficult to find. A "mixing" construction lets us combine polytopes to build new regular and chiral polytopes. By using the chirality group of a polytope, we are able to give simple criteria for when the mix of two polytopes is chiral.
Given an abstract polytope $\cal P$, its flag graph is the edge-coloured graph whose vertices are the flags of $\cal P$ and the $i$-edges correspond to $i$-adjacent flags. Flag graphs of polytopes are maniplexes. On the other hand, given a maniplex $\cal M$, on can define a poset $\cal P_M$ by means of the non empty intersection of its faces. In this paper we give necessary and sufficient conditions (in terms of graphs) on a maniplex $\cal M$ in order for $\cal P_M$ to be an abstract polytope. Moreover, in such case, we show that $\cal M$ is isomorphic to the flag graph of $\cal P_M$. This in …
For any flag simplicial complex $Θ$ obtained by stellar subdividing the boundary of the cross polytope in edges, we define a flag simplicial complex $Γ(Θ)$ (dependent on the sequence of subdivisions) whose $f$-vector is the $γ$-vector of $Θ$. This proves that the $γ$-vector of any such simplicial complex satisfies the Frankl-Füredi-Kalai inequalities, partially solving a conjecture by Nevo and Petersen \cite{np}. We show that when $Θ$ is the dual simplicial complex to a nestohedron, and the sequence of subdivisions corresponds to a flag ordering as defined in \cite{ai}, that $Γ(Θ)$ is equal to…
The full root polytope of type $A$ is the convex hull of all pairwise differences of the standard basis vectors which we represent by forward and backward arrows. We completely classify all flag triangulations of this polytope that are uniform in the sense that the edges may be described as a function of the relative order of the indices of the four basis vectors involved. These fifteen triangulations fall naturally into three classes: three in the lex class, three in the revlex class and nine in the Simion class. We also consider a refined face count where we distinguish between forward and b…
The orbit graph of a k-orbit polytope is a graph on k nodes that shows how the flag orbits are related by flag adjacency. Using orbit graphs, we classify k-orbit polytopes and determine when a k-orbit polytope is i-transitive. We then provide an explicit classification of three-orbit polytopes, and we describe a generating set for their automorphism groups.
What is the minimal closed cone containing all $f$-vectors of cubical $d$-polytopes? We construct cubical polytopes showing that this cone, expressed in the cubical $g$-vector coordinates, contains the nonnegative $g$-orthant, thus verifying one direction of the Cubical Generalized Lower Bound Conjecture of Babson, Billera and Chan. Our polytopes also show that a natural cubical analogue of the simplicial Generalized Lower Bound Theorem does not hold.
A chiral polytope with Schläfli symbol $\{p_1, \ldots, p_{n-1}\}$ has at least $2p_1 \cdots p_{n-1}$ flags, and it is called \emph{tight} if the number of flags meets this lower bound. The Schläfli symbols of tight chiral polyhedra were classified in an earlier paper, and another paper proved that there are no tight chiral $n$-polytopes with $n \geq 6$. Here we prove that there are no tight chiral $5$-polytopes, describe 11 families of tight chiral $4$-polytopes, and show that every tight chiral $4$-polytope covers a polytope from one of those families.
An abstract polytope of rank n is said to be chiral if its automorphism group has precisely two orbits on the flags, such that adjacent flags belong to distinct orbits. The present paper describes a general method for deriving new finite chiral polytopes from old finite chiral polytopes of the same rank. In particular, the technique is used to construct many new examples in ranks 3, 4 and 5.
Marked chain-order polytopes are convex polytopes constructed from a marked poset, which give a discrete family relating a marked order polytope with a marked chain polytope. In this paper, we consider the Gelfand-Tsetlin poset of type A, and realize the associated marked chain-order polytopes as Newton-Okounkov bodies of the flag variety. Our realization connects previous realizations of Gelfand-Tsetlin polytopes and Feigin-Fourier-Littelmann-Vinberg polytopes as Newton-Okounkov bodies in a uniform way. As an application, we prove that the flag variety degenerates into the irreducible normal …
For a polytope we define the {\em flag polynomial}, a polynomial in commuting variables related to the well-known flag vector and describe how to express the the flag polynomial of the Minkowski sum of $k$ standard simplices in a direct and canonical way in terms of the {\em $k$-th master polytope} $P(k)$ where $k\in\nats$. The flag polynomial facilitates many direct computations. To demonstrate this we provide two examples; we first derive a formula for the $f$-polynomial and the maximum number of $d$-dimensional faces of the Minkowski sum of two simplices. We then compute the maximum discrep…
Bisztriczky defines a multiplex as a generalization of a simplex, and an ordinary polytope as a generalization of a cyclic polytope. This paper presents results concerning the combinatorics of multiplexes and ordinary polytopes. The flag vector of the multiplex is computed, and shown to equal the flag vector of a many-folded pyramid over a polygon. Multiplexes, but not other ordinary polytopes, are shown to be elementary. It is shown that all complete subgraphs of the graph of a multiplex determine faces of the multiplex. The toric h-vectors of the ordinary 5-dimensional polytopes are given. G…
A simplicial complex $Δ$ is called flag if all minimal nonfaces of $Δ$ have at most two elements. The following are proved: First, if $Δ$ is a flag simplicial pseudomanifold of dimension $d-1$, then the graph of $Δ$ (i) is $(2d-2)$-vertex-connected and (ii) has a subgraph which is a subdivision of the graph of the $d$-dimensional cross-polytope. Second, the $h$-vector of a flag simplicial homology sphere $Δ$ of dimension $d-1$ is minimized when $Δ$ is the boundary complex of the $d$-dimensional cross-polytope.
Grünbaum, Barnette, and Reay in 1974 completed the characterization of the pairs $(f_i,f_j)$ of face numbers of $4$-dimensional polytopes. Here we obtain a complete characterization of the pairs of flag numbers $(f_0,f_{03})$ for $4$-polytopes. Furthermore, we describe the pairs of face numbers $(f_0,f_{d-1})$ for $d$-polytopes; this description is complete for even $d\ge6$ except for finitely many exceptional pairs that are "small" in a well-defined sense, while for odd $d$ we show that there are also "large" exceptional pairs. Our proofs rely on the insight that "small" pairs need to be defi…
We present a first example of a flag vector of a polyhedral sphere that is not the flag vector of any polytope. Namely, there is a unique 3-sphere with the parameters $(f_0,f_1,f_2,f_3;f_{02})=(12,40,40,12;120)$, but this sphere is not realizable by a convex 4-polytope. The 3-sphere, which is 2-simple and 2-simplicial, was found by Werner (2009); we present results of a computer enumeration which imply that the sphere with these parameters is unique. We prove that it is non-polytopal in two ways: First, we show that it has no oriented matroid, and thus it is not realizable; this proof was foun…
In this paper we study the operation of cutting off edges of a simple $3$-polytope $P$ along the graph $Γ$. We give the criterion when the resulting polytope is simple and when it is flag. As a corollary we prove the analog of Eberhard's theorem about the realization of polygon vectors of simple $3$-polytopes for flag polytopes.
We consider geometric and computational measures of complexity for sets of integer vectors, asking for a qualitative difference between $f$-vectors of simplicial and general $d$-polytopes, as well as flag $f$-vectors of $d$-polytopes and regular CW $(d-1)$-spheres, for $d\ge 4$.
We construct a family of flat semitoric degenerations for the Hibi variety of every finite distributive lattice. The irreducible components of each degeneration are the toric varieties associated with polytopes forming a regular subdivision of the order polytope of the underlying poset. These components are themselves Hibi varieties. For each degeneration in our family we also define the corresponding weight polytope and embed the degeneration into the associated toric variety as the union of orbit closures given by a set of faces. Every such weight polytope projects onto the order polytope wi…
De Loera et al. 2009, showed that when the rank is fixed the Ehrhart polynomial of a matroid polytope can be computed in polynomial time when the number of elements varies. A key to proving this is the fact that the number of simplicial cones in any triangulation of a tangent cone is bounded polynomially in the number of elements when the rank is fixed. The authors speculated whether or not the Ehrhart polynomial could be computed in polynomial time in terms of the number of bases, where the number of elements and rank are allowed to vary. We show here that for the uniform matroid of rank $r$ …
We investigate weighted floating bodies of polytopes. We show that the weighted volume depends on the complete flags of the polytope. This connection is obtained by introducing flag simplices, which translate between the metric and combinatorial structure. Our results are applied in spherical and hyperbolic space. This leads to new asymptotic results for polytopes in these spaces. We also provide explicit examples of spherical and hyperbolic convex bodies whose floating bodies behave completely different from any convex body in Euclidean space.
We prove an upper bound of the form $2^{O(d^2 \mathrm{polylog}\,d)}$ on the number of affine (resp. linear) equivalence classes of, by increasing order of generality, 2-level d-polytopes, d-cones and d-configurations. This in particular answers positively a conjecture of Bohn et al. on 2-level polytopes. We obtain our upper bound by relating affine (resp. linear) equivalence classes of 2-level d-polytopes, d-cones and d-configurations to faces of the correlation cone. We complement this with a $2^{Ω(d^2)}$ lower bound, by estimating the number of nonequivalent stable set polytopes of bipartite…
In this paper, we develop a tropical analog of the classical flag variety that we call the flag Dressian. We find relations, which we call "tropical incidence relations", for when one tropical linear space is contained in another, and show that the flag Dressian is a tropical prevariety. In the case of 2-step flag Dressians, which we call "tropical incidence prevarieties", we find an equivalence between points in this space and induced subdivisions of a hypersimplex, generalizing two parts of an equivalence given by D. Speyer for tropical linear spaces. We attempt to generalize the third part …
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.
The closed cone of flag vectors of Eulerian partially ordered sets is studied. It is completely determined up through rank seven. Half-Eulerian posets are defined. Certain limit posets of Billera and Hetyei are half-Eulerian; they give rise to extreme rays of the cone for Eulerian posets. A new family of linear inequalities valid for flag vectors of Eulerian posets is given.
We introduce the flag-approximability of a convex body to measure how easy it is to approximate by polytopes. We show that the flag-approximability is exactly half the volume entropy of the Hilbert geometry on the body, and that both quantities are maximized when the convex body is a Euclidean ball. We also compute explicitly the asymptotic volume of a convex polytope, which allows us to prove that simplices have the least asymptotic volume.
The {\em hypermetric cone} is defined as the cone of semimetrics satisfying the {\em hypermetric inequalities}. Every {\em Delaunay polytope} corresponds to a ray of this polyhedral cone. The Delaunay polytopes, which correspond to extreme rays are called {\em extreme}. We use this polyhedral cone and the {\em closest vector problem} to present a new technique that allow to find, from a given extreme Delaunay polytope, some new ones. Then, we show some examples of applications of this technique in low-dimensions.
This paper defines, for each graph $G$, a flag vector $fG$. The flag vectors of the graphs on $n$ vertices span a space whose dimension is $p(n)$, the number of partitions on $n$. The analogy with convex polytopes indicates that the linear inequalities satisfied by $fG$ may be both interesting and accessible. Such would provide inequalities both sharp and subtle on the combinatorial structure of $G$. These may be related to Ramsey theory.
McMullen's g-vector is important for simple convex polytopes. This paper postulates axioms for its extension to general convex polytopes. It also conjectures that, for each dimension d, a stated finite calculation gives the formula for the extended g-vector. This calculation is done by computer for d=5 and the results analysed. The conjectures imply new linear inequalities on convex polytope flag vectors. Underlying the axioms is a hypothesised higher-order homology extension to middle perversity intersection homology (order-zero homology), which measures the failure of lower-order homology to…
In this short communication, we observe that the Graphical Traveling Salesman Polyhedron is the intersection of the positive orthant with the Minkowski sum of the Symmetric Traveling Salesman Polytope and the polar of the metric cone. This follows almost trivially from known facts. There are two reasons why we find this observation worth communicating none-the-less: It is very surprising; it helps to understand the relationship between these two important families of polyhedra.
Every polyhedral cone can be described either by its facets or by its extreme rays. Computation of one description from the other is a problem that can be very complex, i.e. one encounter the combinatorial explosion. We present here several methods which allow us to use symmetries for the computation of extreme rays and facets of cones. I give an algorithm for computing adjacencies using a linear programming approach. I give a complete description of the method of incidence, adjacency decomposition method and of the subcone method used for computing with symmetric cones.
Stanley introduced a lattice polytope $\mathcal{C}_P$ arising from a finite poset $P$, which is called the chain polytope of $P$. The geometric structure of $\mathcal{C}_P$ has good relations with the combinatorial structure of $P$. In particular, the Ehrhart polynomial of $\mathcal{C}_P$ is given by the order polynomial of $P$. In the present paper, associated to $P$, we introduce a lattice polytope $\mathcal{E}_{P}$, which is called the enriched chain polytope of $P$, and investigate geometric and combinatorial properties of this polytope. By virtue of the algebraic technique on Gröbner base…
A partially ordered set is r-thick if every nonempty open interval contains at least r elements. This paper studies the flag vectors of graded, r-thick posets and shows the smallest convex cone containing them is isomorphic to the cone of flag vectors of all graded posets. It also defines a k-analogue of the Mobius function and k-Eulerian posets, which are 2k-thick. Several characterizations of k-Eulerian posets are given. The generalized Dehn-Sommerville equations are proved for flag vectors of k-Eulerian posets. A new inequality is proved to be valid and sharp for rank 8 Eulerian posets.
Minkowski sums of simplices in ${\mathbb{R}}^n$ form an interesting class of polytopes that seem to emerge in various situations. In this paper we discuss the Minkowski sum of the simplices $Δ_{k-1}$ in ${\mathbb{R}}^n$ where $k$ and $n$ are fixed, their flags and some of their face lattice structure. In particular, we derive a closed formula for their {\em exponential generating flag function}. These polytopes are simple, include both the simplex $Δ_{n-1}$ and the permutahedron $Π_{n-1}$, and form a Minkowski basis for more general permutahedra.
Through tropical normal idempotent matrices, we introduce isocanted alcoved polytopes, computing their $f$--vectors and checking the validity of the following five conjectures: Bárány, unimodality, $3^d$, 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 $2^{d+1}-2$ vertices, its face lattice is the lattice of proper subsets of $[d+1]$ and its diameter is $d+1$. They are realizati…
The closure of the convex cone generated by all flag $f$-vectors of graded posets is shown to be polyhedral. In particular, we give the facet inequalities to the polar cone of all nonnegative chain-enumeration functionals on this class of posets. These are in one-to-one correspondence with antichains of intervals on the set of ranks and thus are counted by Catalan numbers. Furthermore, we prove that the convolution operation introduced by Kalai assigns extreme rays to pairs of extreme rays in most cases. We describe the strongest possible inequalities for graded posets of rank at most 5.
This note defines a complete h-vector for convex polytopes, which extends the already known toric (or mpih) h-vector and has many similar properties. Complete means that it encodes the whole of the flag vector. First we define the concept of a generalised h-vector and state some properties that follow. The toric h-vector is given as an example. We then define a complete generalised h-vector, and again state properties. Finally, we show that this complete h-vector and all with similar properties will sometimes have negative coefficients. Most of the proofs, and further investigations, will appe…
We study the combinatorics of pseudoline arrangements and their relation to the geometry of flag and Schubert varieties. We associate to each pseudoline arrangement two polyhedral cones, defined in a dual manner. We prove that one of them is the weighted string cone by Littelmann and Berenstein-Zelevinsky. For the other we show how it arises in the framework of cluster varieties and mirror symmetry by Gross-Hacking-Keel-Kontsevich: for the flag variety the cone is the tropicalization of their superpotential while for Schubert varieties a restriction of the superpotential is necessary. We prove…
We construct a 2-parameter family of 4-dimensional polytopes with extreme combinatorial structure: In this family, the ``fatness'' of the f-vector gets arbitrarily close to 9, the ``complexity'' (given by the flag vector) gets arbitrarily close to 16. The polytopes are obtained from suitable deformed products of even polygons by a projection to four-space.
