We consider the problem of constructing an abstract $(n+1)$-polytope $Q$ with $k$ facets isomorphic to a given $n$-polytope $P$, where $k \geq 3$. In particular, we consider the case where we want $Q$ to be $(n-2,n)$-flat, meaning that every $(n-2)$-face is incident to every $n$-face (facet). We show that if $P$ admits such a flat extension for a given $k$, then the facet graph of $P$ is $(k-1)$-colorable. Conversely, we show that if the facet graph is $(k-1)$-colorable and $k-1$ is prime, then $P$ admits a flat extension for that $k$. We also show that if $P$ is facet-bipartite, then for every even $k$, there is a flat extension $P|k$ such that every automorphism of $P$ extends to an automorphism of $P|k$. Finally, if $P$ is a facet-bipartite $n$-polytope and $Q$ is a vertex-bipartite $m$-polytope, we describe a flat amalgamation of $P$ and $Q$, an $(m+n-1)$-polytope that is $(n-2,n)$-flat, with $n$-faces isomorphic to $P$ and co-$(n-2)$-faces isomorphic to $Q$.
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.
Given two convex polytopes, the join, the cartesian product and the direct sum of them are well understood. In this paper we extend these three kinds of products to abstract polytopes and introduce a new product, called the topological product, which also arises in a natural way. We show that these products have unique prime factorization theorems. We use this to compute the automorphism group of a product in terms of the automorphism groups of the factors and show that (non trivial) products are almost never regular or two-orbit polytopes. We finish the paper by studying the monodromy group o…
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…
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…
A $k$-orbit maniplex is one that has $k$ orbits of flags under the action of its automorphism group. In this paper we extend the notion of symmetry type graphs of maps to that of maniplexes and polytopes and make use of them to study $k$-orbit maniplexes, as well as fully-transitive 3-maniplexes. In particular, we show that there are no fully-transtive $k$-orbit 3-mainplexes with $k > 1$ an odd number, we classify 3-orbit mainplexes and determine all face transitivities for 3- and 4-orbit maniplexes. Moreover, we give generators of the automorphism group of a polytope or a maniplex, given its …
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.
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.
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.
Given an abstract $n$-polytope $\mathcal{K}$, an abstract $(n+1)$-polytope $\mathcal{P}$ is an extension of $\mathcal{K}$ if all the facets of $\mathcal{P}$ are isomorphic to $\mathcal{K}$. A chiral polytope is a polytope with maximal rotational symmetry that does not admit any reflections. If $\mathcal{P}$ is a chiral extension of $\mathcal{K}$, then all but the last entry of the Schläfli symbol of $\mathcal{P}$ are determined. In this paper we introduce some constructions of chiral extensions $\mathcal{P}$ of certain chiral polytopes in such a way that the last entry of the Schläfli symbol o…
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.
The mixing operation for abstract polytopes gives a natural way to construct the minimal common cover of two polytopes. In this paper, we apply this construction to the regular convex polytopes, determining when the mix is again a polytope, and completely determining the structure of the mix in each case.
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…
We prove that every finite group is the automorphism group of a finite abstract polytope isomorphic to a face-to-face tessellation of a sphere by topological copies of convex polytopes. We also show that this abstract polytope may be realized as a convex polytope.
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.
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.
The paper gives a collection of open problems on abstract polytopes that were either presented at the Polytopes Day in Calgary or motivated by discussions at the preceding Workshop on Convex and Abstract Polytopes at the Banff International Research Station in May 2005.
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…
The paper investigates connections between abstract polytopes and properly edge colored graphs. Given any finite n-edge-colored n-regular graph G, we associate to G a simple abstract polytope P_G of rank n, called the colorful polytope of G, with 1-skeleton isomorphic to G. We investigate the interplay between the geometric, combinatorial, or algebraic properties of the polytope P_G and the combinatorial or algebraic structure of the underlying graph G, focussing in particular on aspects of symmetry. Several such families of colorful polytopes are studied including examples derived from a Cayl…
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.
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 $2p_i$.
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.
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 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 …
Blind and Mani (1987) proved that the entire combinatorial structure (the vertex-facet incidences) of a simple convex polytope is determined by its abstract graph. Their proof is not constructive. Kalai (1988) found a short, elegant, and algorithmic proof of that result. However, his algorithm has always exponential running time. We show that the problem to reconstruct the vertex-facet incidences of a simple polytope P from its graph can be formulated as a combinatorial optimization problem that is strongly dual to the problem of finding an abstract objective function on P (i.e., a shelling or…
A generalization of incidence relations in abstract polytope has been explored, and parameterized surfaces are used as primers. The abstract orientable incidence structure is defined as an algebraic model of incidence relations, in which some algebraic properties in abtract polytope theory are generalized. The geometric interpretation of abstract orientable incidence structure are also discussed. The orientable incidence structure in a semi-regular normal CW complex are briefly investigated.
A positroid is a special case of a realizable matroid that arose from the study of the totally nonnegative part of the Grassmannian by Postnikov. In this paper, we study the facets of its matroid polytope and the independent set polytope. This allows one to describe the bases and independent sets directly from the decorated permutation, bypassing the use of the Grassmann necklace. We also describe a criterion for determining whether a given cyclic interval is a flat or not using the decorated permutation, then show how it applies to checking the concordancy of positroids.
The maximal rank of an abstract regular polytope for M24, the Mathieu group of degree 24, is 5. There are four such polytopes of rank 5 and in this note we describe them using Curtis's MOG. This description is then used to give an upper bound for the diameter of the chamber graphs of these polytopes.
Given a graph G, the graph associahedron is a simple convex polytope whose face poset is based on the connected subgraphs of G. With the additional assignment of a color palette, we define the colorful graph associahedron, show it to be a collection of simple abstract polytopes, and explore its properties.
We call the $δ$-vector of an integral convex polytope of dimension $d$ flat if the $δ$-vector is of the form $(1,0,\ldots,0,a,\ldots,a,0,\ldots,0)$, where $a \geq 1$. In this paper, we give the complete characterization of possible flat $δ$-vectors. Moreover, for an integral convex polytope $\mathcal{P} \subset \mathbb{R}^N$ of dimension $d$, we let $i(\mathcal{P},n)=|n\mathcal{P} \cap \mathbb{Z}^N|$ and $\ i^*(\mathcal{P},n)=|n(\mathcal{P} \setminus \partial \mathcal{P}) \cap \mathbb{Z}^N|.$ By this characterization, we show that for any $d \geq 1$ and for any $k,\ell \geq 0$ with $k+\ell \le…
A matroid is a machine capturing linearity of mathematical objects and producing combinatorial structures. Matroid structure arises everywhere since linearity is a ubiquitous concept. One natural way to obtain matroids is by considering hyperplane arrangements, which give rise to convex polytopes called matroid polytopes. Much research has been conducted on these three areas: matroids, matroid polytopes, and hyperplane arrangements. However, substantial gaps in our knowledge remain, and the correspondence diagram between those areas needs to be more extensive. For instance, currently, there is…
There are two main thrusts in the theory of regular and chiral polytopes: the abstract, purely combinatorial aspect, and the geometric one of realizations. This brief survey concentrates on the latter. The dimension of a faithful realization of a finite abstract regular polytope in some euclidean space is no smaller than its rank, while that of a chiral polytope must strictly exceed the rank. There are similar restrictions on the dimensions of realizations of regular and chiral apeirotopes. From the viewpoint of realizations in a fixed dimension, the problems are now completely solved in up to…
Neighborly polytopes are those that maximize the number of faces in each dimension among all polytopes with the same number of vertices. Despite their extremal properties they form a surprisingly rich class of polytopes, which has been widely studied and is the subject of many open problems and conjectures. In this paper, we study the enumeration of neighborly polytopes beyond the cases that have been computed so far. To this end, we enumerate neighborly oriented matroids --- a combinatorial abstraction of neighborly polytopes --- of small rank and corank. In particular, if we denote by OM($r,…
This paper presents a technique for constructing new chiral or regular polyhedra (or maps) from self-dual abstract chiral polytopes of rank 4. From improperly self-dual chiral polytopes we derive "Petrie-Coxeter-type" polyhedra (abstract chiral analogues of the classical Petrie-Coxeter polyhedra) and investigate their groups of automorphisms.
The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a self-contained introduction to matroid polytopes, we present a geometric construction of the Bergman fan, and we discuss its relationship with the simplicial complex of nested sets in the lattice of flats. The Bergman complex is triangulated by the nested set complex, and the two complexes coincide if and only if every connected flat remains connected after contracting along any subflat. This sharpens a result of Ardila-Klivans who showed that the Bergman c…
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.
The multiplihedra {M_n} form a family of polytopes originating in the study of higher categories and homotopy theory. While the multiplihedra may be unfamiliar to the algebraic combinatorics community, it is nestled between two families of polytopes that certainly are not: the permutahedra {S_n} and associahedra {Y_n}. The maps between these families reveal several new Hopf structures on tree-like objects nestled between the Malvenuto-Reutenauer (MR) Hopf algebra of permutations and the Loday-Ronco (LR) Hopf algebra of planar binary trees. We begin their study here, constructing a module over …
We prove that for every complex classical group $G$ the string polytope associated to a special reduced decomposition and any dominant integral weight $λ$ will be a lattice polytope if and only if the highest weight representation of the Lie algebra of $G$ with highest weight $λ$ integrates to a representation of $G$ itself. This affirms an earlier conjecture and shows that every partial flag variety of a complex classical group admits a flat projective degeneration to a Gorenstein Fano toric variety.
A k-system of the graph G(P) of a simple polytope P is a set of induced subgraphs of G(P) that shares certain properties with the set of subgraphs induced by the k-faces of P. This new concept leads to polynomial-size certificates in terms of G(P) for both the set of vertex sets of facets as well as for abstract objective functions (AOF) in the sense of Kalai. Moreover, it is proved that an acyclic orientation yields an AOF if and only if it induces a unique sink on every 2-face.
The combinatorial structure of a d-dimensional simple convex polytope can be reconstructed from its abstract graph [Blind & Mani 1987, Kalai 1988]. However, no polynomial/efficient algorithm is known for this task, although a polynomially checkable certificate for the correct reconstruction was found by [Joswig, Kaibel & Koerner 2000]. A much stronger certificate would be given by the following characterization of the facet subgraphs, conjectured by M. Perles: ``The facet subgraphs of the graph of a simple d-polytope are exactly all the (d-1)-regular, connected, induced, non-separating subgrap…
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…
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…
This paper contains a classication of the regular minimal abstract polytopes that act as covers for the convex polyhedral prisms and antiprisms. It includes a detailed discussion of their topological structure, and completes the enumeration of such covers for convex uniform polyhedra. Additionally, this paper addresses related structural questions in the theory of string C-groups.
The physical information encoded in the cosmological late-time wavefunction of the universe is tied to its singularity structure and its behaviour as such singularities are approached. One important singularity is identified by the vanishing of the total energy, where the wavefunction reduces to the physics of scattering in flat space. In this paper, we discuss the behaviour of the perturbative wavefunction as its other singularities are approached and the role played by the flat-space scattering, in the simplified context of the class of toy models admitting a first principle definition in te…
We study a family of posets and the associated chain and order polytopes. We identify the order polytope as a maximal Kogan face in a Gelfand-Tsetlin polytope of a multiple of a fundamental weight. We show that the character of such a Kogan face equals to the character of a Demazure module which occurs in the irreducible representation of $\mathfrak{sl}_n$ having highest weight multiple of fundamental weight and for any such Demazure module there exists a corresponding poset and associated maximal Kogan face. We prove that the chain polytope parametrizes a monomial basis of the associated PBW-…
The Monotone Upper Bound Problem asks for the maximal number M(d,n) of vertices on a strictly-increasing edge-path on a simple d-polytope with n facets. More specifically, it asks whether the upper bound M(d,n)<=M_{ubt}(d,n) provided by McMullen's (1970) Upper Bound Theorem is tight, where M_{ubt}(d,n) is the number of vertices of a dual-to-cyclic d-polytope with n facets. It was recently shown that the upper bound M(d,n)<=M_{ubt}(d,n) holds with equality for small dimensions (d<=4: Pfeifle, 2003) and for small corank (n<=d+2: Gärtner et al., 2001). Here we prove that it is not tight in genera…
We study the abstract regular polyhedra with automorphism groups that act faithfully on their vertices, and show that each non-flat abstract regular polyhedron covers a "vertex-faithful" polyhedron with the same number of vertices. We then use this result and earlier work on flat polyhedra to study abstract regular polyhedra based on the size of their vertex set. In particular, we classify all regular polyhedra where the number of vertices is prime or twice a prime. We also construct the smallest regular polyhedra with a prime squared number of vertices.
