Polytope Research Articles

Smooth polytopes with negative Ehrhart coefficients

We present examples of smooth lattice polytopes in dimensions 3 and higher with the maximal possible number of negative Ehrhart coefficients. This answers a question by Bruns. We also discuss Berline–Vergne valuations as a useful tool in proving Ehrhart positivity results.

Packing convex homothetic polytopes into a cuboid

Packing convex homothetic polytopes into a cuboid

A geometrical upper bound on the inflaton range

We argue that in type IIB LVS string models, after including the leading order moduli stabilisation effects, the moduli space for the remaining flat directions is compact due the Calabi-Yau Kähler cone conditions. In cosmological applications, this gives an inflaton field range which is bounded from above, in analogy with recent results from the weak gravity and swampland conjectures. We support our claim by explicitly showing that it holds for all LVS vacua with h1,1 = 3 obtained from 4-dimensional reflexive polytopes. In particular, we first search for all Calabi-Yau threefolds from the Kreuzer-Skarke list with h1,1 = 2, 3 and 4 which allow for LVS vacua, finding several new LVS geometries which were so far unknown. We then focus on the h1,1 = 3 cases and show that the Kähler cones of all toric hypersurface threefolds force the effective 1-dimensional LVS moduli space to be compact. We find that the moduli space size can generically be trans-Planckian only for K3 fibred examples.

Small $f$-vectors of 3-spheres and of 4-polytopes

We present a new algorithmic approach that can be used to determine whether a given quadruple $(f_0,f_1,f_2,f_3)$ is the f-vector of any convex 4-dimensional polytope. By implementing this approach, we classify the f-vectors of 4-polytopes in the range $f_0+f_3\le22$. In particular, we thus prove that there are f-vectors of cellular 3-spheres with the intersection property that are not f-vectors of any convex 4-polytopes, thus answering a question that may be traced back to the works of Steinitz (1906/1922). In the range $f_0+f_3\le22$, there are exactly three such f-vectors with $f_0\le f_3$, namely $(10,32,33,11)$, $(10,33,35,12)$, and $(11,35,35,11)$.

The L2-torsion polytope of amenable groups

The L2-torsion polytope of amenable groups

Structure of 2-skeletons of higher dimensional regular polytopes

Structure of 2-skeletons of higher dimensional regular polytopes

Improved bounds on the diameter of lattice polytopes

We show that the largest possible diameter $\delta(d,k)$ of a $d$-dimensional polytope whose vertices have integer coordinates ranging between $0$ and $k$ is at most $kd-\lceil2d/3\rceil$ when $k\geq3$. In addition, we show that $\delta(4,3)=8$. This substantiates the conjecture whereby $\delta(d,k)$ is at most $\lfloor(k+1)d/2\rfloor$ and is achieved by a Minkowski sum of lattice vectors.

Topology of moment-angle manifolds arising from flag nestohedra

We construct a family of manifolds, one for each $n\geq 2$, having a nontrivial Massey $n$-product in their cohomology. These manifolds turn out to be smooth closed 2-connected manifolds with a compact torus action called moment-angle manifolds $\mathcal Z_P$, whose orbit spaces are simple $n$-dimensional polytopes $P$ obtained from a $n$-cube by a sequence of truncations of faces of codimension 2 only (2-truncated cubes). Moreover, the polytopes $P$ are flag nestohedra but not graph-associahedra. We compute some bigraded Betti numbers $\beta^{-i,2(i+1)}(Q)$ for an associahedron $Q$ in terms of its graph structure and relate it to the structure of the loop homology (Pontryagin algebra) $H_{*}(\Omega\mathcal Z_Q)$. We also study triple Massey products in $H^{*}(\mathcal Z_Q)$ for a graph-associahedron $Q$.

Constructions of families of three-dimensional polytopes, characteristic patches of fullerenes, and Pogorelov polytopes

We describe the combinatorics of three families of simple - dimensional polytopes which play an important role in various problems of algebraic topology, hyperbolic geometry, graph theory, and their applications. The first family consists of simple polytopes with at most hexagonal faces. The second family consists of Pogorelov polytopes. The third family consists of fullerenes and is the intersection of the first two. We show that in the case of fullerenes there are stronger results than for the first two. Our main tools are -belts of faces, simple partitions of a disc and the operations of transformation and connected sum.

Simple polytopes without small separators

We show that by cutting off the vertices and then the edges of neighborly cubical polytopes, one obtains simple 4-dimensional polytopes with n vertices such that all separators of the graph have size at least Ω(n/log3/2 n). This disproves a conjecture by Kalai from 1991/2004.

Global spectra, polytopes and stacky invariants

Given a convex polytope, we define its geometric spectrum, a stacky version of Batyrev’s stringy E-functions, and we prove a stacky version of a formula of Libgober and Wood about the E-polynomial of a smooth projective variety. As an application, we get a closed formula for the variance of the geometric spectrum and a Noether’s formula for two dimensional Fano polytopes (polytopes whose vertices are primitive lattice points; a Fano polytope is not necessarily smooth). We also show that this geometric spectrum is equal to the algebraic spectrum (the spectrum at infinity of a tame Laurent polynomial whose Newton polytope is the polytope alluded to). This gives an explanation and some positive answers to Hertling’s conjecture about the variance of the spectrum of tame regular functions.

Diagonal Complexes for Punctured Polygons

It is known that taken together, all collections of nonintersecting diagonals in a convex planar n-gon give rise to a (combinatorial type of a) convex (n − 3)-dimensional polytope As n called the Stasheff polytope, or associahedron. In the paper, we act in a similar way by taking a convex planar n-gon with k labeled punctures. All collections of mutually nonintersecting and mutually nonhomotopic topological diagonals yield a complex As n,k . We prove that it is a topological ball. We also show a natural cellular fibration Asn,k → As n,k−1 . A special example is delivered by the case k = 1. Here the vertices of the complex are labeled by all possible permutations together with all possible bracketings on n distinct entries. This hints to a relationship with M. Kapranov’s permutoassociahedron.

Interior regularity for the generalized Abreu equation

We study the generalized Abreu equation in [Formula: see text]-dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform [Formula: see text]-stability.

Cohomological rigidity of manifolds defined by 3-dimensional polytopes

A family of closed manifolds is said to be cohomologically rigid if a cohomology ring isomorphism implies a diffeomorphism for any two manifolds in the family. Cohomological rigidity is established here for large families of 3-dimensional and 6-dimensional manifolds defined by 3-dimensional polytopes. The class of 3-dimensional combinatorial simple polytopes different from tetrahedra and without facets forming 3- and 4-belts is studied. This class includes mathematical fullerenes, that is, simple 3- polytopes with only 5-gonal and 6-gonal facets. By a theorem of Pogorelov, any polytope in admits in Lobachevsky 3-space a right-angled realisation which is unique up to isometry. Our families of smooth manifolds are associated with polytopes in the class . The first family consists of 3-dimensional small covers of polytopes in , or equivalently, hyperbolic 3-manifolds of Löbell type. The second family consists of 6-dimensional quasitoric manifolds over polytopes in . Our main result is that both families are cohomologically rigid, that is, two manifolds and from either family are diffeomorphic if and only if their cohomology rings are isomorphic. It is also proved that if and are diffeomorphic, then their corresponding polytopes and are combinatorially equivalent. These results are intertwined with classical subjects in geometry and topology such as the combinatorics of 3-polytopes, the Four Colour Theorem, aspherical manifolds, a diffeomorphism classification of 6-manifolds, and invariance of Pontryagin classes. The proofs use techniques of toric topology. Bibliography: 69 titles.

Antiprismlessness, or: Reducing Combinatorial Equivalence to Projective Equivalence in Realizability Problems for Polytopes

This article exhibits a 4-dimensional combinatorial polytope that has no antiprism, answering a question posed by Bernt Lindstrom. As a consequence, any realization of this combinatorial polytope has a face that it cannot rest upon without toppling over. To this end, we provide a general method for solving a broad class of realizability problems. Specifically, we show that for any first-order semialgebraic property that faces inherit, the given property holds for some realization of every combinatorial polytope if and only if the property holds from some projective copy of every polytope. The proof uses the following result by Below. Given any polytope with vertices having algebraic coordinates, there is a combinatorial "stamp" polytope with a specified face that is projectively equivalent to the given polytope in all realizations.

On the Real-Rootedness of the Veronese Construction for Rational Formal Power Series

We study real sequences $\{a_{n}\}_{n\in \mathbb{N}}$ that eventually agree with a polynomial. We show that if the numerator polynomial of its rational generating series is of degree $s$ and has only nonnegative coefficients, then the numerator polynomial of the subsequence $\{ a_{rn+i}\}_{n\in \mathbb{N}}$, $0\leq i<r$, has only nonpositive, real roots for all $r\geq s-i$. We apply our results to combinatorially positive valuations on polytopes and to Hilbert functions of Veronese submodules of graded Cohen-Macaulay algebras. In particular, we prove that the Ehrhart $h^\ast$-polynomial of the $r$-th dilate of a $d$-dimensional polytope has only distinct, negative, real roots if $r\geq \min \{s+1,d\}$. This proves a conjecture of Beck and Stapledon (2010).

Describing polyhedral tilings and higher dimensional polytopes by sequence of their two-dimensional components

Polyhedral tilings are often used to represent structures such as atoms in materials, grains in crystals, foams, galaxies in the universe, etc. In the previous paper, we have developed a theory to convert a way of how polyhedra are arranged to form a polyhedral tiling into a codeword (series of numbers) from which the original structure can be recovered. The previous theory is based on the idea of forming a polyhedral tiling by gluing together polyhedra face to face. In this paper, we show that the codeword contains redundant digits not needed for recovering the original structure, and develop a theory to reduce the redundancy. For this purpose, instead of polyhedra, we regard two-dimensional regions shared by faces of adjacent polyhedra as building blocks of a polyhedral tiling. Using the present method, the same information is represented by a shorter codeword whose length is reduced by up to the half of the original one. Shorter codewords are easier to handle for both humans and computers, and thus more useful to describe polyhedral tilings. By generalizing the idea of assembling two-dimensional components to higher dimensional polytopes, we develop a unified theory to represent polyhedral tilings and polytopes of different dimensions in the same light.

Cost of seven-brane gauge symmetry in a quadrillion F-theory compactifications

We study seven-branes in $O(10^{15})$ four-dimensional F-theory compactifications where seven-brane moduli must be tuned in order to achieve non-abelian gauge symmetry. The associated compact spaces $B$ are the set of all smooth weak Fano toric threefolds. By a study of fine star regular triangulations of three dimensional reflexive polytopes, the number of such spaces is estimated to be $5.8\times 10^{14}\lesssim N_\text{bases}\lesssim 1.8\times 10^{17}$. Typically hundreds or thousands of moduli must be tuned to achieve symmetry for $h^{11}(B)<10$, but the average number drops sharply into the range $O(25)$-$O(200)$ as $h^{11}(B)$ increases. For some low rank groups, such as $SU(2)$ and $SU(3)$, there exist examples where only a few moduli must be tuned in order to achieve seven-brane gauge symmetry.

Cohomological rigidity of manifolds defined by 3-dimensional polytopes

Семейство замкнутых многообразий называется когомологически жeстким, если изоморфизм колец когомологий влечeт диффеоморфизм для любых двух многообразий из этого семейства. В центре внимания обзора - результаты о когомологической жeсткости для широких семейств шестимерных и трeхмерных многообразий, задаваемых трeхмерными многогранниками. Рассматривается класс $\mathscr{P}$ трeхмерных комбинаторных простых многогранников $P$, отличных от тетраэдра, грани которых не образуют $3$- и $4$-поясов. Этот класс содержит все математические фуллерены, т. е. простые трeхмерные многогранники, имеющие лишь пятиугольные и шестиугольные грани. Согласно теореме Погорелова, многогранник из класса $\mathscr{P}$ допускает прямоугольную реализацию в пространстве Лобачевского, которая единственна с точностью до изометрии. Изучаемые семейства гладких многообразий ассоциированы с многогранниками из класса $\mathscr{P}$. Первое семейство составляют трeхмерные малые накрытия над многогранниками из $\mathscr{P}$ или, эквивалентно, гиперболические 3-многообразия типа Лeбелля. Второе семейство состоит из шестимерных квазиторических многообразий над многогранниками из $\mathscr{P}$. Наш основной результат заключается в том, что оба эти семейства являются когомологически жeсткими, т. е. два многообразия $M$ и $M'$ из любого из этих семейств диффеоморфны тогда и только тогда, когда изоморфны их кольца когомологий. Более того, доказывается, что если $M$ и $M'$ диффеоморфны, то соответствующие многогранники $P$ и $P'$ комбинаторно эквивалентны. Эти результаты переплетаются с классическими сюжетами геометрии и топологии, которые составили обзорную часть нашей статьи. Речь идeт о комбинаторике трeхмерных многогранников, теореме о четырeх красках, асферических многообразиях, классификации гладких шестимерных многообразий и инвариантности классов Понтрягина. Доказательства в основной части статьи используют технику торической топологии. Библиография: 68 названий.

The stringy Euler number of Calabi–Yau hypersurfaces in toric varieties and the Mavlyutov duality

We show that minimal models of nondegenerated hypersufaces defined by Laurent polynomials with a $d$-dimensional Newton polytope $\Delta$ are Calabi-Yau varieties $X$ if and only if the Fine interior of $\Delta$ consists of a single lattice point. We give a combinatorial formula for computing the stringy Euler number of $X$. This formula allows to test mirror symmetry in cases when $\Delta$ is not a reflexive polytope. In particular we apply this formula to pairs of lattice polytopes $(\Delta, \Delta^{\vee})$ that appear in the Mavlyutov's generalization of the polar duality for reflexive polytopes. Some examples of Mavlyutov's dual pairs $(\Delta, \Delta^{\vee})$ show that the stringy Euler numbers of the corresponding Calabi-Yau varieties $X$ and $X^{\vee}$ may not satisfy the expected topological mirror symmetry test: $e_{\rm st}(X) = (-1)^{d-1} e_{\rm st}(X^{\vee})$. This shows the necessity of an additional condition on Mavlyutov's pairs $(\Delta, \Delta^\vee)$.