Lattice Polytopes Research Articles

A Genetic Algorithm to Search the Space of Ehrhart h*-Vectors

We describe a genetic algorithm to find candidates for h * -vectors satisfying given properties in the space of integers vectors of finite length. We use an implementation of such algorithm to find a 52-dimensional lattice polytope having a non-unimodal h * -vector which is the Cartesian product of two lattice polytopes having unimodal h * -vectors. This counterexample answers negatively to a question by Ferroni and Higashitani.

Combinatorial and algebraic mutations of toric Fano 3-folds and mass deformations of 2d(0,2) quiver gauge theories

We argue that algebraic and combinatorial polytope mutations of Fano 3-folds can be identified with mass deformations of associated 2d (0,2) supersymmetric gauge theories realized by brane brick models. These are type IIA brane configurations that realize a large family of 2d worldvolume theories on probe D1-branes at toric Calabi-Yau 4-folds. We show that brane brick models that are related by mass deformations associated to algebraic and combinatorial polytope mutations of Fano 3-folds have mesonic moduli spaces with the same number of generators. We show that mesonic flavor charges of these generators form convex reflexive lattice polytopes that are dual to the toric diagrams of the Fano 3-folds. The generating function of mesonic gauge invariant operators, also known as the Hilbert series of the mesonic moduli space, appears to be identical for such brane brick models under a particular refinement originating from the U(1)R charges in the brane brick model following the mass deformation. Published by the American Physical Society 2024

Faces of cosmological polytopes

A cosmological polytope is a lattice polytope introduced by Arkani-Hamed, Benincasa, and Postnikov in their study of the wavefunction of the universe in a class of cosmological models. More concretely, they construct a cosmological polytope for any Feynman diagram, i.e., an undirected graph. In this paper, we initiate a combinatorial study of these polytopes. We give a complete description of their faces, identify minimal faces that are not simplices and compute the number of faces in specific instances. In particular, we give a recursive description of the f -vector of cosmological polytopes of trees.

Classification of Width 1 Lattice Tetrahedra by Their Multi-Width

AbstractWe introduce the multi-width of a lattice polytope and use this to classify and count all lattice tetrahedra with multi-width $$(1,w_2,w_3)$$ ( 1 , w 2 , w 3 ) . The approach used in this classification can be extended into a computer algorithm to classify lattice tetrahedra of any given multi-width. We use this to classify tetrahedra with multi-width $$(2,w_2,w_3)$$ ( 2 , w 2 , w 3 ) for small $$w_2$$ w 2 and $$w_3$$ w 3 and make conjectures about the function counting lattice tetrahedra of any multi-width.

A new invariant of lattice polytopes

The maximal degree of monomials belonging to the unique minimal system of monomial generators of the canonical module $\omega (K[\mathcal{P}])$ of the toric ring $K[\mathcal{P}]$ defined by a lattice polytope $\mathcal{P}$ will be studied. It is shown that if $\mathcal{P}$ possesses an interior lattice point, then the maximal degree is at most $\dim \mathcal{P} - 1$, and that this bound is the best possible in general.

On the finite generation of valuation semigroups on toric surfaces

We provide a combinatorial criterion for the finite generation of a valuation semigroup associated with an ample divisor on a smooth toric surface and a non-toric valuation of maximal rank. As an application, we construct a lattice polytope such that none of the valuation semigroups of the associated polarized toric variety coming from one-parameter subgroups and centered at a non-toric point are finitely generated.

Perfectly matchable set polynomials and h⁎-polynomials for stable set polytopes of complements of graphs

A subset S of vertices of a graph G is called a perfectly matchable set of G if the subgraph induced by S contains a perfect matching. The perfectly matchable set polynomial of G, first made explicit by Ohsugi and Tsuchiya, is the (ordinary) generating function p(G;z) for the number of perfectly matchable sets of G.In this work, we provide explicit recurrences for computing p(G;z) for an arbitrary (simple) graph and use these to compute the Ehrhart h⁎-polynomials for certain lattice polytopes. Namely, we show that p(G;z) is the h⁎-polynomial for certain classes of stable set polytopes, whose vertices correspond to stable sets of G.

Integer Decomposition Property of Polytopes

We study the integer decomposition property of lattice polytopes associated with the $n$-dimensional smooth complete fans with at most $n+3$ rays. Using the classification of smooth complete fans by Kleinschmidt and Batyrev and a reduction to lower dimensional polytopes we prove the integer decomposition property for lattice polytopes in this setting.

Nearly Gorenstein Polytopes

In this paper, we study nearly Gorensteinness of Ehrhart rings arising from lattice polytopes. We give necessary conditions and sufficient conditions on lattice polytopes for their Ehrhart rings to be nearly Gorenstein. Using this, we give an efficient method for constructing nearly Gorenstein polytopes. Moreover, we determine the structure of nearly Gorenstein (0, 1)-polytopes and characterise nearly Gorensteinness of edge polytopes and graphic matroids.

Facets of Random Symmetric Edge Polytopes, Degree Sequences, and Clustering

Symmetric edge polytopes are lattice polytopes associated with finite simple graphs that are of interest in both theory and applications. We investigate the facet structure of symmetric edge polytopes for various models of random graphs. For an Erd\H{o}s-Renyi random graph, we identify a threshold probability at which with high probability the symmetric edge polytope shares many facet-supporting hyperplanes with that of a complete graph. We also investigate the relationship between the average local clustering, also known as the Watts-Strogatz clustering coefficient, and the number of facets for graphs with either a fixed number of edges or a fixed degree sequence. We use well-known Markov Chain Monte Carlo sampling methods to generate empirical evidence that for a fixed degree sequence, higher average local clustering in a connected graph corresponds to higher facet numbers in the associated symmetric edge polytope.

Castelnuovo Polytopes

It is known that the sectional genus of a polarized variety has an upper bound, which is an extension of the Castelnuovo bound on the genus of a projective curve. Polarized varieties whose sectional genus achieve this bound are called Castelnuovo. On the other hand, a lattice polytope is called Castelnuovo if the associated polarized toric variety is Castelnuovo. Kawaguchi characterized Castelnuovo polytopes having interior lattice points in terms of their h∗-vectors. In this paper, as a generalization of this result, we present a characterization of all Castelnuovo polytopes. Finally, as an application of our characterization, we give a sufficient criterion for a lattice polytope to be IDP.

Deep Lattice Points in Zonotopes, Lonely Runners, and Lonely Rabbits

Abstract Let $P \subseteq {\mathbb {R}}^{d}$ be a polytope and let $\textbf {w}$ be an interior point of $P$. The coefficient of asymmetry$\operatorname {ca}(P,\textbf {w}):= \min \{ \lambda \geq 1: \textbf {w} - P \subseteq \lambda (P - \textbf {w}) \}$ of $P$ about $\textbf {w}$ has been studied extensively in the realm of Hensley’s conjecture on the maximal volume of a $d$-dimensional lattice polytope that contains a fixed positive number of interior lattice points. We zero in on the coefficient of asymmetry for lattice zonotopes, that is, Minkowski sums of line segments with integer endpoints. Our main result gives the existence of an interior lattice point for which the coefficient of asymmetry is bounded above by an explicit constant in $\Theta (d \log \log d)$, for any lattice zonotope that has an interior lattice point. Our work is both inspired by and feeds on Wills’ lonely runner conjecture from Diophantine approximation: we make intensive use of a discrete version of this conjecture (which, in fact, has been proved), and reciprocally, we reformulate the lonely runner conjecture in terms of the coefficient of asymmetry for certain lattice zonotopes.

Proof of a Conjecture of Batyrev and Juny on Gorenstein Polytopes

AbstractA d-dimensional lattice polytope P is Gorenstein if it has a multiple rP that is a reflexive polytope up to translation by a lattice vector. The difference $$d+1-r$$ d + 1 - r is called the degree of P. We show that a Gorenstein polytope is a lattice pyramid if its dimension is at least three times its degree. This was previously conjectured by Batyrev and Juny. We also present a refined conjecture and prove it for IDP Gorenstein polytopes.

Cayley Sums and Minkowski Sums of Lattice Polytopes

Cayley Sums and Minkowski Sums of Lattice Polytopes

Some Computational Study of the Root Distribution of Ehrhart Polynomials

In this paper, we investigate the root distribution of the Ehrhart polynomials of lattice polytopes. When the lattice polytope is reflexive, the roots of the Ehrhart polynomial distribute symmetrically with respect to the line [Formula: see text]. A special case of this distribution is when all the roots lie on this line. Our main concern is to find out which reflexive polytopes satisfy this special condition. Such lattice polytopes are called CL-polytopes. Another special case opposite to this is when all the roots are real. Such lattice polytopes are called real polytopes. The first topic of this paper is the Ehrhart polynomials of equatorial spheres, which are related to graded posets. We discuss the CL-ness of the equatorial Ehrhart polynomials for complete graded posets and zig-zag posets. The second topic is to investigate the relation of the root distribution of the Ehrhart polynomials of a reflexive polytope [Formula: see text] and its dual [Formula: see text]. We discuss the CL-ness and realness of [Formula: see text] and [Formula: see text] in pair, of dimensions up to 4. Throughout this paper, we investigate these problems by computer calculation.

The equivariant Ehrhart theory of polytopes with order-two symmetries

We study the equivariant Ehrhart theory of families of polytopes that are invariant under a non-trivial action of the group with order two. We study families of polytopes whose equivariant H ∗ H^\ast -polynomial both succeed and fail to be effective, in particular, the symmetric edge polytopes of cycle graphs and the rational cross-polytope. The latter provides a counterexample to the effectiveness conjecture if the requirement that the vertices of the polytope have integral coordinates is loosened to allow rational coordinates. Moreover, we exhibit such a counterexample whose Ehrhart function has period one and coincides with the Ehrhart function of a lattice polytope.

Affine subspace concentration conditions for centered polytopes

AbstractRecently, K.‐Y. Wu introduced affine subspace concentration conditions for the cone volumes of polytopes and proved that the cone volumes of centered, reflexive, smooth lattice polytopes satisfy these conditions. We extend the result to arbitrary centered polytopes.

Sandpile Solitons in Higher Dimensions

Let $$p\in {\mathbb {Z}}^n$$ be a primitive vector and $$\Psi :{\mathbb {Z}}^n\rightarrow {\mathbb {Z}}, z\rightarrow \min (p\cdot z, 0)$$ . The theory of husking allows us to prove that there exists a pointwise minimal function among all integer-valued superharmonic functions equal to $$\Psi $$ “at infinity”. We apply this result to sandpile models on $${\mathbb {Z}}^n$$ . We prove existence of so-called solitons in a sandpile model, discovered in 2-dim setting by S. Caracciolo, G. Paoletti, and A. Sportiello and studied by the author and M. Shkolnikov in previous papers. We prove that, similarly to 2-dim case, sandpile states, defined using our husking procedure, move changeless when we apply the sandpile wave operator (that is why we call them solitons). We prove an analogous result for each lattice polytope A without lattice points except its vertices. Namely, for each function $$\begin{aligned} \Psi :{\mathbb {Z}}^n\rightarrow {\mathbb {Z}}, z\rightarrow \min _{p\in A\cap {\mathbb {Z}}^n}(p\cdot z+c_p), c_p\in {\mathbb {Z}}\end{aligned}$$ there exists a pointwise minimal function among all integer-valued superharmonic functions coinciding with $$\Psi $$ “at infinity”. The Laplacian of the latter function corresponds to what we observe when solitons, corresponding to the edges of A, intersect (see Fig. 1).

PQ-Type Adjacency Polytopes of Join Graphs

PQ-type adjacency polytopes nabla ^PQ _G are lattice polytopes arising from finite graphs G. There is a connection between nabla ^PQ _G and the engineering problem known as power-flow study, which models the balance of electric power on a network of power generation. In particular, the normalized volume of nabla ^PQ _G plays a central role. In the present paper, we focus on the case where G is a join graph. In particular, formulas of the h^*-polynomial and the normalized volume of nabla ^PQ _G of a join graph G are presented. Moreover, we give explicit formulas of the h^*-polynomial and the normalized volume of nabla ^PQ _G when G is a complete multipartite graph or a wheel graph.

Lattice zonotopes of degree 2

The Ehrhart polynomial{text {ehr}}_P (n) of a lattice polytope P gives the number of integer lattice points in the n-th dilate of P for all integers nge 0. The degree of P is defined as the degree of its h^*-polynomial, a particular transformation of the Ehrhart polynomial with many useful properties which serves as an important tool for classification questions in Ehrhart theory. A zonotope is the Minkowski (pointwise) sum of line segments. We classify all Ehrhart polynomials of lattice zonotopes of degree 2 thereby complementing results of Scott (Bull Aust Math Soc 15(3), 395–399, 1976), Treutlein (J Combin Theory Ser A 117(3), 354–360, 2010), and Henk and Tagami (Eur J Combin 30(1), 70–83, 2009). Our proof is constructive: by considering solid-angles and the lattice width, we provide a characterization of all 3-dimensional zonotopes of degree 2.