Lattice Polygons Research Articles

The volume polynomial of lattice polygons

We prove that every indefinite quadratic form with non-negative integer coefficients is the volume polynomial of a pair of lattice polygons. This solves the discrete version of the Heine–Shephard problem for two bodies in the plane. As an application, we show how to construct a pair of planar tropical curves (or a pair of divisors on a toric surface) with given intersection number and self-intersection numbers.

A first proof of knot localization for polymers in a nanochannel

Abstract Based on polymer scaling theory and numerical evidence, Orlandini, Tesi, Janse van Rensburg and Whittington conjectured in 1996 that the limiting entropy of knot-type K lattice polygons is the same as that for unknot polygons, and that the entropic critical exponent increases by one for each prime knot in the knot decomposition of K. This Knot Entropy (KE) conjecture is consistent with the idea that for unconfined polymers, knots occur in a localized way (the knotted part is relatively small compared to polymer length). For full confinement (to a sphere or box), numerical evidence suggests that knots are much less localized. Numerical evidence for nanochannel or tube confinement is mixed, depending on how the size of a knot is measured. Here we outline the proof that the KE conjecture holds for polygons in the ∞ x 2 x 1 lattice tube and show that knotting is localized when a connected-sum measure of knot size is used. Similar results are established for linked polygons. This is the first model for which the knot entropy conjecture has been proved.

Lattice Structures Built with Different Polygon Hollow Shapes: A Review on Their Analytical Modelling and Engineering Applications

Lattice structures are useful in the aerospace, automotive, infrastructural, and medical fields due to the way they incorporate a lightweight design and good mechanical properties, because of their hollow shapes. This review paper documents work carried out using various analytical models for lattice structures designed with different polygon hollow shapes, for loading in the in-plane and out-of-plane directions, in order to advise their ranking in terms of mechanical behaviour. A primer on lattice structures and polygon hollow shapes is first provided. This is followed by a review of relevant analytical models applied to lattice structures with various polygon hollow shapes that are available in the literature, and then a ranking of the polygon hollow structures in terms of their mechanical properties is performed. Following on from this, a review of the mechanical properties of polygon hollow structures is given. Engineering applications of different polygon hollow structures are then identified. A next-generation structural optimisation and design guide is then highlighted, and some of the primary prospective areas to be focused on when designing lattice parts are pointed out. The last section highlights current challenges, as well as recommendations for extending the use of design for the additive manufacturing of lattice parts.

Bounds on Area Involving Lattice Size

The lattice size of a lattice polygon $P$ is a combinatorial invariant of $P$ that was recently introduced in relation to the problem of bounding the total degree and the bi-degree of the defining equation of an algebraic curve. In this paper, we establish sharp lower bounds on the area of plane convex bodies $P\subset\mathbb{R}^2$ that involve the lattice size of $P$. In particular, we improve bounds given by Arnold, and Bárány and Pach. We also provide a classification of minimal lattice polygons $P\subset\mathbb{R}^2$ of fixed lattice size ${\operatorname{ls_\square}}(P)$.

Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices

We have analysed the recently extended series for the number of self-avoiding walks (SAWs) that cross an L × L square between diagonally opposed corners. The number of such walks is known to grow as We have made more precise the estimate of based on additional series coefficients provided by several authors, and refined analysis techniques. We estimate that We have also studied the subdominant behaviour, and conjecture that CL(1)∼λSL2+bL+c⋅Lg, where and We implemented a very efficient algorithm for enumerating paths on the square and hexagonal lattices making use of a minimal perfect hash function and in-place memory updating of the arrays for the counts of the number of paths. Using this algorithm we extended and then analysed series for SAWs spanning the square lattice and self-avoiding polygons (SAPs) crossing the square lattice. These are known to also grow as The sub-dominant term λ b is found to be the same as for SAWs crossing the square, while the exponent for spanning SAWs and for SAPs. We have also studied the analogous problems on the hexagonal lattice, and generated series for a number of geometries. In particular, we study SAWs and SAPs crossing rhomboidal, triangular and square domains on the hexagonal lattice, as well as SAWs spanning a rhombus. We estimate that the analogous growth constant so an even more precise estimate than found for the square lattice. We also give estimates of the sub-dominant terms.

Congruences for Consecutive Coefficients of Gaussian Polynomials with Crank Statistics


 
 
 In this paper, we establish infinite families of congruences in consecutive arithmetic progressions modulo any odd prime $\ell$ for the function $p\big(n,m,N\big)$, which enumerates the partitions of $n$ into at most $m$ parts with no part larger than $N$. We also treat the function $p\big(n,m,(a,b]\big)$, which bounds the largest part above and below, and obtain similar infinite families of congruences.
 For $m \leq 4$ and $\ell = 3$, simple combinatorial statistics called "cranks" witness these congruences. We prove this analytically for $m=4$, and then both analytically and combinatorially for $m = 3$. Our combinatorial proof relies upon explicit dissections of convex lattice polygons.
 
 For $m \leq 4$ and $\ell = 3$, simple combinatorial statistics called ``cranks" witness these congruences. We prove this analytically for $m=4$, and then both analytically and combinatorially for $m = 3$. Our combinatorial proof relies upon explicit dissections of convex lattice polygons. 
 
 

Deformations of Dimer Models

The combinatorial mutation of polygons, which transforms a given lattice polygon into another one, is an important operation to understand mirror partners for two-dimensional Fano manifolds, and the mutation-equivalent polygons give ${\mathbb Q}$-Gorenstein deformation-equivalent toric varieties. On the other hand, for a dimer model, which is a bipartite graph described on the real two-torus, one can assign a lattice polygon called the perfect matching polygon. It is known that for each lattice polygon $P$ there exists a dimer model having $P$ as the perfect matching polygon and satisfying certain consistency conditions. Moreover, a dimer model has rich information regarding toric geometry associated with the perfect matching polygon. In this paper, we introduce a set of operations which we call deformations of consistent dimer models, and show that the deformations of consistent dimer models realize the combinatorial mutations of the associated perfect matching polygons.

The moduli space of tropical curves with fixed Newton polygon

Abstract Given a lattice polygon, we study the moduli space of all tropical plane curves with that Newton polygon. We determine a formula for the dimension of this space in terms of combinatorial properties of that polygon. If this polygon is nonhyperelliptic or maximal and hyperelliptic, then this formula matches the dimension of the moduli space of nondegenerate algebraic curves with the given Newton polygon.

Large deviations of convex polyominoes

Many open problems in machine learning, pattern recognition, and geometric analysis require enumeration of different types of lattice polygons, and in particular convex polyominoes. In this work, we develop a large deviation principle for convex polyominoes under different restrictions, such as fixed area and/or perimeter.

Moduli dimensions of lattice polygons

Given a lattice polygon P with g interior lattice points, we can associate to \(P\) two moduli spaces: the moduli space of algebraic curves that are non-degenerate with respect to \(P\) and the moduli space of tropical curves of genus g with Newton polygon P. We completely classify the possible dimensions such a moduli space can have in the tropical case. For non-hyperelliptic polygons, the dimension must be between g and \(2g+1\) and can take on any integer value in this range, with exceptions only in the cases of genus 3, 4, and 7. We provide a similar result for hyperelliptic polygons, for which the range of dimensions is from g to \(2g-1\). In the case of non-hyperelliptic polygons, our results also hold for the moduli space of algebraic curves that are non-degenerate with respect to P.

Prism graphs in tropical plane curves

Any smooth tropical plane curve contains a distinguished trivalent graph called its skeleton. In 2020 Morrison and Tewari proved that the so-called big face graphs cannot be the skeleta of tropical curves for genus $12$ and greater. In this paper we answer an open question they posed to extend their result to the prism graphs, proving that they are the skeleton of a smooth tropical plane curve precisely when the genus is at most $11$. Our main tool is a classification of lattice polygons with two points than can simultaneously view all others, without having any one point that can observe all others.

Lattice Size and Generalized Basis Reduction in Dimension Three

The lattice size of a lattice polytope P was defined and studied by Schicho, and Castryck and Cools. They provided an “onion skins” algorithm for computing the lattice size of a lattice polygon P in $$\mathbb R^2$$ based on passing successively to the convex hull of the interior lattice points of P. We explain the connection of the lattice size to the successive minima of $$K=(P+(-P))^*$$ and to the lattice reduction with respect to the general norm that corresponds to K. It follows that the generalized Gauss algorithm of Kaib and Schnorr (which is faster than the “onion skins” algorithm) computes the lattice size of any convex body in $$\mathbb R^2$$ . We extend the work of Kaib and Schnorr to dimension three, providing a fast algorithm for lattice reduction with respect to the general norm defined by a convex origin-symmetric body $$K\subset \mathbb R^3$$ . We also explain how to recover the successive minima of K and the lattice size of P from the obtained reduced basis and therefore provide a fast algorithm for computing the lattice size of any convex body $$P\subset \mathbb R^3$$ .

Bounding the Kreuzer‐Skarke Landscape

AbstractWe study Calabi‐Yau threefolds with large Hodge numbers by constructing and counting triangulations of reflexive polytopes. By counting points in the associated secondary polytopes, we show that the number of fine, regular, star triangulations of polytopes in the Kreuzer‐Skarke list is bounded above by . Adapting a result of Anclin on triangulations of lattice polygons, we obtain a bound on the number of triangulations of each 2‐face of each polytope in the list. In this way we prove that the number of topologically inequivalent Calabi‐Yau hypersurfaces arising from the Kreuzer‐Skarke list is bounded above by 10428. We introduce efficient algorithms for constructing representative ensembles of Calabi‐Yau hypersurfaces, including the extremal case , and we study the distributions of topological and physical data therein. Finally, we demonstrate that neural networks can accurately predict these data once the triangulation is encoded in terms of the secondary polytope.

Convex lattice polygons with all lattice points visible

Two lattice points are visible to one another if there exist no other lattice points on the line segment connecting them. In this paper we study convex lattice polygons that contain a lattice point such that all other lattice points in the polygon are visible from it. We completely classify such polygons, show that there are finitely many of lattice width greater than 2, and computationally enumerate them. As an application of this classification, we prove new obstructions to graphs arising as skeleta of tropical plane curves.

Quiver gauge theories: beyond reflexivity

Reflexive polygons have been extensively studied in a variety of contexts in mathematics and physics. We generalize this programme by looking at the 45 different lattice polygons with two interior points up to SL(2,ℤ) equivalence. Each corresponds to some affine toric 3-fold as a cone over a Sasaki-Einstein 5-fold. We study the quiver gauge theories of D3-branes probing these cones, which coincide with the mesonic moduli space. The minimum of the volume function of the Sasaki-Einstein base manifold plays an important role in computing the R-charges. We analyze these minimized volumes with respect to the topological quantities of the compact surfaces constructed from the polygons. Unlike reflexive polytopes, one can have two fans from the two interior points, and hence give rise to two smooth varieties after complete resolutions, leading to an interesting pair of closely related geometries and gauge theories.

The free energy of compressed lattice knots

A compressed knotted ring polymer in a confining cavity is modelled by a knotted lattice polygon confined in a cube in . The GAS algorithm (Janse van Rensburg and Rechnitzer 2011 J. Knot Theol. Raman 20 1145–71) is used to sample lattice polygons of fixed knot type in a confining cube and to estimate the free energy of confined lattice knots. Lattice polygons of knot types the unknot, the trefoil knot, and the figure eight knot, are sampled and the free energies are estimated as functions of the concentration of monomers in the confining cube. The data show that the free energy is a function of knot type at low concentrations, and (mean-field) Flory–Huggins theory (Flory 1942 J. Chem. Phys. 10 51–61; Huggins 1942 J. Phys. Chem. 46 151–8) is used to model the free energy as a function of monomer concentration. The Flory interaction parameter of knotted lattice polygons in is also estimated.

Внешние биллиарды вне правильного десятиугольника: периодичность почти всех орбит и существование апериодической орбиты

Внешние биллиарды были введены Б. Нойманном в 50-х годах ХХ века и стали популярны в 70-х благодаря Ю. Мозеру, который рассматривал внешний, или двойственный, биллиард как игрушечную модель небесной механики. Задача об устойчивости Солнечной системы обладает тем свойством, что "легко выписать n уравнений движения частиц, но сложно понять это движение интуитивно"; в связи с этим, Мозер предложил рассмотреть ранее поставленную Б. Нойманном задачу внешнего биллиарда, обладающую тем же свойством.Одним из классических примеров динамической систем является внешний биллиард вне правильного ????-угольника; в частности, с ним связаны проблемы существования апериодической траектории, а также полноты периодических точек. Эти проблемы решены лишь для ограниченного количества частных случаев.При ???? = 3, 4, 6 стол является решеточным, и, как следствие, апериодических точек нет, а периодические точки образуют множество полной меры. В 1993 году, С. Табачникову удалось найти апериодическую точку в случае правильного пятиугольника; сделано это было с помощью ренормализационной схемы - метода, имеющего фундаментальное значение при исследовании самоподобных динамических систем.По мнению Р. Шварца, следующими по сложности являются случаи n = 10,8,12; в этих случаях, а также в случае ???? = 5 для внешнего биллиарда удается построить ренормализационную схему, которая, как пишет Шварц, “позволяет дать (как минимум, в принципе) полное описание того, что происходит”.Позже, автору удалось обнаружить самоподобные структуры и построить ренормализационную схему для случаев правильных восьми- и двенадцатиугольника.Данная же статья посвящена внешнему биллиарду вне правильного десятиугольника. Доказано существование апериодической орбиты для внешнего биллиарда вне правильного десятиугольника, а также, что почти все траектории такого внешнего биллиарда являются периодическими; явно выписаны все возможные периоды. В основе работы лежит классическая технология поиска и исследования ренормализационной схемы. Возникающие в случае ???? = 10 периодические структуры похожи на периодические структуры в случае ???? = 5, но все же имеют свои особенности.

The free energy of compressed lattice knots

Abstract IOP Publishing has withdrawn this article upon the author's request due to several issues with the mathematical content. The osmotic pressure of monomers in a knotted ring polymer in a confining cavity is modelled by a lattice polygon confined in a cube in ${\mathbb Z}^3$. These polygons can be knotted and are called lattice knots. In this paper the GAS algorithm \cite{JvRR11} is used to estimate the free energy of lattice knots of knot types the unknot, the trefoil knot, and the figure eight knot, as a function of the concentration of monomers in the confining cube. The data show that the free energy is a function of knot type at low concentrations, and (mean-field) Flory-Huggins theory \cite{Flory42,H42} is used to model the free energy as a function of monomer concentration. The Flory interaction parameter of lattice polygons in ${\mathbb Z}^3$ is also estimated. At critical values of the concentration the osmotic pressure may vanish, and these critical concentrations, suitably rescaled, is dependent on knot type.

Computing graded Betti tables of toric surfaces

We present various facts on the graded Betti table of a projectively embedded toric surface, expressed in terms of the combinatorics of its defining lattice polygon. These facts include explicit formulas for a number of entries, as well as a lower bound on the length of the quadratic strand that we conjecture to be sharp (and prove to be so in several special cases). We also present an algorithm for determining the graded Betti table of a given toric surface by explicitly computing its Koszul cohomology and report on an implementation in SageMath. It works well for ambient projective spaces of dimension up to roughly 25 25 , depending on the concrete combinatorics, although the current implementation runs in finite characteristic only. As a main application we obtain the graded Betti table of the Veronese surface ν 6 ( P 2 ) ⊆ P 27 \nu _6(\mathbb {P}^2) \subseteq \mathbb {P}^{27} in characteristic 40 009 40\,009 . This allows us to formulate precise conjectures predicting what certain entries look like in the case of an arbitrary Veronese surface ν d ( P 2 ) \nu _d(\mathbb {P}^2) .

An extremal property of lattice polygons

We find the critical number of vertices of a convex lattice polygon that guarantees that the polygon contains at least one point of a given square sublattice. As a tool, we prove Diophantine inequalities relating the number of edges of a broken line and the coordinates of its endpoints.