We study $γ$-vectors associated with $h^*$-vectors of symmetric edge polytopes both from a deterministic and a probabilistic point of view. On the deterministic side, we prove nonnegativity of $γ_2$ for any graph and completely characterize the case when $γ_2 = 0$. The latter also confirms a conjecture by Lutz and Nevo in the realm of symmetric edge polytopes. On the probabilistic side, we show that the $γ$-vectors of symmetric edge polytopes of most Erdős-Rényi random graphs are asymptotically almost surely nonnegative up to any fixed entry. This proves that Gal's conjecture holds asymptotically almost surely for arbitrary unimodular triangulations in this setting.
This paper is a survey of recent advances as well as open problems in the study of face numbers of centrally symmetric simplicial polytopes and spheres. The topics discussed range from neighborliness of centrally symmetric polytopes and the upper bound theorem for centrally symmetric simplicial spheres to the generalized lower bound theorem for centrally symmetric simplicial polytopes and the lower bound conjecture for centrally symmetric simplicial spheres and manifolds.
Stanley proved that for any centrally symmetric simplicial $d$-polytope $P$ with $d\geq 3$, $g_2(P) \geq {d \choose 2}-d$. We provide a characterization of centrally symmetric $d$-polytopes with $d\geq 4$ that satisfy this inequality as equality. This gives a natural generalization of the classical Lower Bound Theorem for simplicial polytopes to the setting of centrally symmetric simplicial polytopes.
Using a ribbon structure of the graph, we construct a dissection of the symmetric edge polytope of a graph into unimodular simplices. Our dissection is shellable, and one can interpret the elements of the resulting $h$-vector via graph theory. This gives an elementary method for computing the $h^*$-vector of the symmetric edge polytope.
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…
Symmetric edge polytopes, a.k.a. PV-type adjacency 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 study of the algebraic Kuramoto equations of unmixed form whose Newton polytopes are the symmetric edge polytopes. The interplay between the geometric structure of symmetric edge polytopes and the topological structure of the underlying graphs has been a recurring theme in recent studies. In particular, ``facet/face su…
A $δ$-vector $δ(\Pc)= (δ_0, δ_1, ..., δ_d)$ is called shifted symmetric if $δ_{d-i} = δ_{i+1}$ for each $0 \leq i \leq [(d-1)/2]$. A natural family of $(0,1)$-polytopes with shifted symmetric $δ$-vectors will be studied.
We show that there exist k-neighborly centrally symmetric d-dimensional polytopes with 2(n+d) vertices, where k(d,n)=Theta(d/(1+log ((d+n)/d))). We also show that this bound is tight.
In Ehrhart theory, the $h^*$-vector of a rational polytope often provide insights into properties of the polytope that may be otherwise obscured. As an example, the Birkhoff polytope, also known as the polytope of real doubly-stochastic matrices, has a unimodal $h^*$-vector, but when even small modifications are made to the polytope, the same property can be very difficult to prove. In this paper, we examine the $h^*$-vectors of a class of polytopes containing real doubly-stochastic symmetric matrices.
What is the maximum number of vertices that a centrally symmetric 2-neighborly polytope of dimension $d$ can have? It is known that the answer does not exceed $2^d$. Here we provide an explicit construction showing that it is at least $2^{d-1}+2$.
Symmetric Grothendieck polynomials are inhomogeneous versions of Schur polynomials that arise in combinatorial $K$-theory. A polynomial has saturated Newton polytope (SNP) if every lattice point in the polytope is an exponent vector. We show Newton polytopes of these Grothendieck polynomials and their homogeneous components have SNP. Moreover, the Newton polytope of each homogeneous component is a permutahedron. This addresses recent conjectures of C. Monical-N. Tokcan-A. Yong and of A. Fink-K. Mészáros-A. St. Dizier in this special case.
Generalized alcoved polytopes are polytopes whose facet normals are roots in a given root system. We call a set of points in an alcoved polytope a generating set if there does not exist a strictly smaller alcoved polytope containing it. The type $A$ alcoved polytopes are precisely the tropical polytopes that are also convex in the usual sense. In this case the tropical generators form a generating set. We show that for any root system other than $F_4$, every alcoved polytope invariant under the natural Weyl group action has a generating set of cardinality equal to the Coxeter number of the roo…
Given a family of lattice polytopes, two common questions in Ehrhart Theory are determining when a polytope has the integer decomposition property and determining when a polytope is reflexive. While these properties are of independent interest, the confluence of these properties is a source of active investigation due to conjectures regarding the unimodality of the $h^\ast$-polynomial. In this paper, we consider the Newton polytopes arising from two families of polynomials in algebraic combinatorics: Schur polynomials and inflated symmetric Grothendieck polynomials. In both cases, we prove tha…
The aim of the paper is to calculate face numbers of simple generalized permutohedra, and study their f-, h- and gamma-vectors. These polytopes include permutohedra, associahedra, graph-associahedra, simple graphic zonotopes, nestohedra, and other interesting polytopes. We give several explicit formulas for h-vectors and gamma-vectors involving descent statistics. This includes a combinatorial interpretation for gamma-vectors of a large class of generalized permutohedra which are flag simple polytopes, and confirms for them Gal's conjecture on nonnegativity of gamma-vectors. We calculate expli…
A reflexive polytope, respectively its associated Gorenstein toric Fano variety, is called pseudo-symmetric, if the polytope has a centrally symmetric pair of facets. Here we present a complete classification of pseudo-symmetric simplicial reflexive polytopes together with some applications. This generalizes a result of Ewald on pseudo-symmetric nonsingular toric Fano varieties.
Symmetric edge polytopes are a class of lattice polytopes constructed from finite simple graphs. In the present paper we highlight their connections to the Kuramoto synchronization model in physics -- where they are called adjacency polytopes -- and to Kantorovich--Rubinstein polytopes from finite metric space theory. Each of these connections motivates the study of symmetric edge polytopes of particular classes of graphs. We focus on such classes and apply algebraic-combinatorial methods to investigate invariants of the associated symmetric edge polytopes.
The asymptotic volume of the polytope of symmetric stochastic matrices can be determined by asymptotic enumeration techniques as in the case of the Birkhoff polytope. These methods can be extended to polytopes of symmetric stochastic matrices with given diagonal, if this diagonal varies not too wildly. To this end, the asymptotic number of symmetric matrices with natural entries, zero diagonal and varying row sums is determined and a third order correction factor to this is examined.
We investigate convex polytopes of doubly stochastic matrices having special structures: symmetric, Hankel symmetric, centrosymmetric, and both symmetric and Hankel symmetric. We determine dimensions of these polytopes and classify their extreme points. We also determine a basis of the real vector spaces generated by permutation matrices with these special structures.
We analyze a remarkable class of centrally symmetric polytopes, the Hansen polytopes of split graphs. We confirm Kalai's 3^d-conjecture for such polytopes (they all have at least 3^d nonempty faces) and show that the Hanner polytopes among them (which have exactly 3^d nonempty faces) correspond to threshold graphs. Our study produces a new family of Hansen polytopes that have only 3^d+16 nonempty faces.
We study the problem of covering R^d by overlapping translates of a convex body P, such that almost every point of R^d is covered exactly k times. Such a covering of Euclidean space by translations is called a k-tiling. The investigation of tilings (i.e. 1-tilings in this context) by translations began with the work of Fedorov and Minkowski. Here we extend the investigations of Minkowski to k-tilings by proving that if a convex body k-tiles R^d by translations, then it is centrally symmetric, and its facets are also centrally symmetric. These are the analogues of Minkowski's conditions for 1-t…
In a previous paper, we showed how to use the Ehrhart function $L_P(s)$, defined by $L_P(s) = \#(sP \cap \mathbb Z^d)$, to reconstruct a polytope $P$. More specifically, we showed that, for rational polytopes $P$ and $Q$, if $L_{P + w}(s) = L_{Q + w}(s)$ for all integer vectors $w$, then $P = Q$. In this paper we show the same result, but assuming that $P$ and $Q$ are symmetric convex bodies instead of rational polytopes.
We give an explicit construction, based on Hadamard matrices, for an infinite series of floor{sqrt{d}/2}-neighborly centrally symmetric d-dimensional polytopes with 4d vertices. This appears to be the best explicit version yet of a recent probabilistic result due to Linial and Novik, who proved the existence of such polytopes with a neighborliness of d/400.
In a previous article, we proved tight lower bounds for the coefficients of the generalized $h$-vector of a centrally symmetric rational polytope using intersection cohomology of the associated projective toric variety. Here we present a new proof based on the theory of combinatorial intersection cohomology developed by Barthel, Brasselet, Fieseler and Kaup. This theory is also valid for nonrational polytopes when there are no corresponding toric varieties. So we can establish our bounds for centrally symmetric polytopes even without requiring them to be rational.
In 1997 Oda conjectured that every smooth lattice polytope has the integer decomposition property. We prove Oda's conjecture for centrally symmetric $3$-dimensional polytopes, by showing they are covered by lattice parallelepipeds and unimodular simplices.
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}…
In this paper we use the technique of Hopf algebras and quasi-symmetric functions to study the combinatorial polytopes. Consider the free abelian group $\mathcal{P}$ generated by all combinatorial polytopes. There are two natural bilinear operations on this group defined by a direct product $\times $ and a join $\divideontimes$ of polytopes. $(\mathcal{P},\times)$ is a commutative associative bigraded ring of polynomials, and $\mathcal{RP}=(\mathbb Z\varnothing\oplus\mathcal{P},\divideontimes)$ is a commutative associative threegraded ring of polynomials. The ring $\mathcal{RP}$ has the struct…
We present explicit constructions of centrally symmetric polytopes with many faces: first, we construct a d-dimensional centrally symmetric polytope P with about (1.316)^d vertices such that every pair of non-antipodal vertices of P spans an edge of P, second, for an integer k>1, we construct a d-dimensional centrally symmetric polytope P of an arbitrarily high dimension d and with an arbitrarily large number N of vertices such that for some 0 < delta_k < 1 at least (1-delta_k^d) {N choose k} k-subsets of the set of vertices span faces of P, and third, for an integer k>1 and a>0, we construct …
We show that the $γ$-vector of the order complex of any polytope is the f-vector of a balanced simplicial complex. This is done by proving this statement for a subclass of Stanley's S-shellable spheres which includes all polytopes. The proof shows that certain parts of the cd-index, when specializing $c=1$ and considering the resulted polynomial in $d$, are the f-polynomials of simplicial complexes that can be colored with "few" colors. We conjecture that the cd-index of a regular CW-sphere is itself the flag f-vector of a colored simplicial complex in a certain sense.
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 $\left\{{p_1 \atop p_1},\ldots,p_{n-1}\right\}$.
We consider 1-skeletons of the symmetric and asymmetric traveling salesperson polytopes whose vertices are all possible Hamiltonian tours in the complete directed or undirected graph, and the edges are geometric edges or one-dimensional faces of the polytope. It is known that the question whether two vertices of the symmetric or asymmetric traveling salesperson polytopes are nonadjacent is NP-complete. A sufficient condition for nonadjacency can be formulated as a combinatorial problem: if from the edges of two Hamiltonian tours we can construct two complementary Hamiltonian tours, then the co…
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.
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.
Symmetric edge polytopes $\mathcal{A}_G$ of type A are lattice polytopes arising from the root system $A_n$ and finite simple graphs $G$. There is a connection between $\mathcal{A}_G$ and the Kuramoto synchronization model in physics. In particular, the normalized volume of $\mathcal {A}_G$ plays a central role. In the present paper, we focus on a particular class of graphs. In fact, for any cactus graph $G$, we give a formula for the $h^*$-polynomial of $\mathcal{A}_{\widehat{G}}$ by using matching generating polynomials, where $\widehat{G}$ is the suspension of $G$. This gives also a formula…
In 1989 Kalai stated the three conjectures A, B, C of increasing strength concerning face numbers of centrally symmetric convex polytopes. The weakest conjecture, A, became known as the ``$3^d$-conjecture''. It is well-known that the three conjectures hold in dimensions d \leq 3. We show that in dimension 4 only conjectures A and B are valid, while conjecture C fails. Furthermore, we show that both conjectures B and C fail in all dimensions d \geq 5.
We present explicit constructions of centrally symmetric 2-neighborly d-dimensional polytopes with about 3^{d/2} = (1.73)^d vertices and of centrally symmetric k-neighborly d-polytopes with about 2^{c_k d} vertices where c_k=3/20 k^2 2^k. Using this result, we construct for a fixed k > 1 and arbitrarily large d and N, a centrally symmetric d-polytope with N vertices that has at least (1-k^2 (gamma_k)^d) binom(N, k) faces of dimension k-1, where gamma_2=1/\sqrt{3} = 0.58 and gamma_k = 2^{-3/{20k^2 2^k}} for k > 2. Another application is a construction of a set of 3^{d/2 -1}-1 points in R^d ever…
In this paper, we consider terminal reflexive polytopes arising from finite directed graphs and study the problem of deciding which directed graphs yield smooth Fano polytopes. We show that any centrally symmetric or pseudo-symmetric smooth Fano polytopes can be obtained from directed graphs. Moreover, by using directed graphs, we provide new examples of smooth Fano polytopes whose corresponding varieties admit Kähler--Einstein metrics.
Graphs (1-skeletons) of Traveling-Salesman-related polytopes have attracted a lot of attention. Pedigree polytopes are extensions of the classical Symmetric Traveling Salesman Problem polytopes (Arthanari 2000) whose graphs contain the TSP polytope graphs as spanning subgraphs (Arthanari 2013). Unlike TSP polytopes, Pedigree polytopes are not "symmetric", e.g., their graphs are not vertex transitive, not even regular. We show that in the graph of the pedigree polytope, the quotient minimum degree over number of vertices tends to 1 as the number of cities tends to infinity.
We report here a computation giving the complete list of facets for the cut polytopes over several very symmetric graphs with $15-30$ edges, including $K_8$, $K_{3,3,3}$, $K_{1,4,4}$, $K_{5,5}$, some other $K_{l,m}$, $K_{1,l,m}$, $Prism_7, APrism_6$, Möbius ladder $M_{14}$, Dodecahedron, Heawood and Petersen graphs. For $K_8$, it shows that the huge lists of facets of the cut polytope $CUTP_8$ and cut cone $CUT_8$, given in [CR] is complete. We also confirm the conjecture that any facet of $CUTP_8$ is adjacent to a triangle facet. The lists of facets for $K_{1,l,m}$ with $(l,m)=(4,4),(3,5),(3,…
