Polyhedral Complex Research Articles

Oriented Closed Polyhedral Maps and the Kitaev Model

A kind of combinatorial map, called arrow presentation, is proposed to encode the data of the oriented closed polyhedral complexes $\Sigma$ on which the Hopf algebraic Kitaev model lives. We develop a theory of arrow presentations which underlines the role of the dual of the double $\mathcal{D}(\Sigma)^*$ of $\Sigma$ as being the Schreier coset graph of the arrow presentation, explains the ribbon structure behind curves on $\mathcal{D}(\Sigma)^*$ and facilitates computation of holonomy with values in the algebra of the Kitaev model. In this way, we can prove ribbon operator identities for arbitrary f.d. C$^*$-Hopf algebras and arbitrary oriented closed polyhedral maps. By means of a combinatorial notion of homotopy designed specially for ribbon curves, we can rigorously formulate ''topological invariance'' of states created by ribbon operators.

Algorithmic reconstruction of the fiber of persistent homology on cell complexes

Let K be a finite simplicial, cubical, delta or CW complex. The persistence map PH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PH}$$\\end{document} takes a filter f:K→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:K\\rightarrow \\mathbb {R}$$\\end{document} as input and returns the barcodes of the sublevel set persistent homology of f in each dimension. We address the inverse problem: given target barcodes D, computing the fiber PH-1(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PH}^{-1}(D)$$\\end{document}. For this, we use the fact that PH-1(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PH}^{-1}(D)$$\\end{document} decomposes as a polyhedral complex when K is a simplicial complex, and we generalise this result to arbitrary based chain complexes. We then design and implement a depth-first search that recovers the polytopes forming the fiber PH-1(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PH}^{-1}(D)$$\\end{document}. As an application, we solve a corpus of 120 sample problems, providing a first insight into the statistical structure of these fibers, for general CW complexes.

Locally Finite Completions of Polyhedral Complexes

Locally Finite Completions of Polyhedral Complexes

Graded local cohomology of modules over semigroup rings

We give a combinatorial description of local cohomology modules of a graded module over a semigroup ring, with support at the graded maximal ideal. This combinatorial framework yields Hochster-type formulas for the Hilbert series of such local cohomology modules in terms of the homology of finitely many polyhedral cell complexes. A Cohen–Macaulay criterion immediately follows. We also provide an alternative proof of a result of [18] characterizing Cohen–Macaulay affine semigroup rings.

Tropical Compactification via Ganter’s Algorithm

AbstractWe describe a canonical compactification of a polyhedral complex in Euclidean space. When the recession cones of the polyhedral complex form a fan, the compactified polyhedral complex is a subspace of a tropical toric variety. In this case, the procedure is analogous to the tropical compactifications of subvarieties of tori. We give an analysis of the combinatorial structure of the compactification and show that its Hasse diagram can be computed via Ganter’s algorithm. Our algorithm is implemented in and shipped with .

Topological characteristics of grain boundary networks during severe plastic deformations of copper alloys

The grain boundary character distribution strongly affects the properties of polycrystalline materials. Grain boundaries of similar characters form networks, whose topological invariants can be considered as distribution descriptors. Understanding the evolution of such descriptors during severe plastic deformations (SPD) can elucidate the evolution of properties and underpin the design of processing routes for target behaviour. For topological analysis of grain boundary networks, polycrystalline materials are considered here as polyhedral complexes with grain boundaries classified into two types — low-angle and high-angle. Changes of grain boundary types are calculated using sub-grain rotations, which reflects the physical mechanism of microstructure evolution during SPD. A non-physical approach, by direct conversions of low-angle to high-angle boundaries, is also explored as a reference to demonstrate the impact of the physical constraint imposed by rotations. Reported is the discovery of topological phase transitions in the grain boundary networks which might take place during severe plastic deformations of different copper alloys. Depending on the evolution approach, the transitions correspond to zeros of the Euler characteristic, or of the logarithm of the inverse connectivity, of the grain boundary network. The relations between these transitions and the fraction of high-angle grain boundaries, and between the fraction of high-angle grain boundaries and plastic strain obtainable experimentally, provide new perspectives for grain boundary engineering and network design. Determining the dominant evolution mechanism and critical accumulated strain for a given material and processing route requires further experimental studies of triple junction evolution.

Interconversion of Highly Entangled Polyhedra into Concave Polyhedra by Nitrate‐Induced Ternary Coordination

AbstractEntangled (M3L2)n polyhedral complexes represent a unique class of supramolecular architectures that are stabilized by relatively weak metal–acetylene interactions in cooperation with conventional metal–pyridyl coordination. Counter‐anion exchange of these complexes with a nitrate (NO3−) ion triggered formal metal insertion between the metal centers, and a heteroleptic ternary coordination mode with acetylenic, pyridyl, and nitrate donors was generated on the metal centers. As a result, the main frameworks of the polyhedral complexes M18L12 and M12L8 were formally extended into a new series of concave polyhedra having the compositions M21L12 and M13L8, respectively. This transformation also resulted in the local disconnection of the highly entangled trifurcate topology of the framework, providing clues toward the skeletal editing of extended and complex three‐dimensional (3D) architectures.

Interconversion of Highly Entangled Polyhedra into Concave Polyhedra by Nitrate-Induced Ternary Coordination.

Entangled (M3L2) n polyhedral complexes represent a unique class of supramolecular architectures that are stabilized by relatively weak metal-acetylene interactions in cooperation with conventional metal-pyridyl coordination. Counter-anion exchange of these complexes with a nitrate (NO3-) ion triggered formal metal insertion between the metal centers, and a heteroleptic ternary coordination mode with acetylenic, pyridyl, and nitrate donors was generated on the metal centers. As a result, the main frameworks of the polyhedral complexes M18L12 and M12L8 were formally extended into a new series of concave polyhedra having the compositions M21L12 and M13L8, respectively. This transformation also resulted in the local disconnection of the highly entangled trifurcate topology of the framework, providing clues toward the skeletal editing of extended and complex three-dimensional (3D) architectures.

Quadratic regularization of bilevel pricing problems and application to electricity retail markets

We consider the profit-maximization problem solved by an electricity retailer who aims at designing a menu of contracts. This is an extension of the unit-demand envy-free pricing problem: customers aim to choose a contract maximizing their utility based on a reservation bill and multiple price coefficients (attributes). A basic approach supposes that the customers have deterministic utilities; then, the response of each customer is highly sensitive to price since it concentrates on the best offer. A second classical approach is to consider logit model to add a probabilistic behavior in the customers’ choices. To circumvent the intrinsic instability of the former and the resolution difficulties of the latter, we introduce a quadratically regularized model of customer’s response, which leads to a quadratic program under complementarity constraints (QPCC). This allows to robustify the deterministic model, while keeping a strong geometrical structure. In particular, we show that the customer’s response is governed by a polyhedral complex, in which every polyhedral cell determines a set of contracts which is effectively chosen. Moreover, the deterministic model is recovered as a limit case of the regularized one. We exploit these geometrical properties to develop a pivoting heuristic, which we compare with implicit or non-linear methods from bilevel programming, showing the effectiveness of the approach. Throughout the paper, the electricity retailer problem is our guideline, and we present a numerical study on this application case.

Homomorphism complexes, reconfiguration, and homotopy for directed graphs

The neighborhood complex of a graph was introduced by Lovász to provide topological lower bounds on chromatic number. More general homomorphism complexes of graphs were further studied by Babson and Kozlov. Such ‘Hom complexes’ are also related to mixings of graph colorings and other reconfiguration problems, as well as a notion of discrete homotopy for graphs. Here we initiate the detailed study of Hom complexes for directed graphs (digraphs). For any pair of digraphs graphs G and H, we consider the polyhedral complex Hom⃗(G,H) that parametrizes the directed graph homomorphisms f:G→H. This construction can be seen as a special case of the poset structure on the set of multihomomorphisms in more general categories, as introduced by Kozlov, Matsushita, and others. Hom complexes of digraphs have applications in the study of chains in graded posets and cellular resolutions of monomial ideals.We study examples of directed Hom⃗ complexes and relate their topological properties to certain graph operations including products, adjunctions, and foldings. We introduce a notion of a neighborhood complex for a digraph and prove that its homotopy type is recovered as the Hom⃗ complex of homomorphisms from a directed edge. We establish a number of results regarding the topology of directed neighborhood complexes, including the dependence on directed bipartite subgraphs, a digraph version of the Mycielski construction, as well as vanishing theorems for higher homology. The Hom⃗ complexes of digraphs provide a natural framework for reconfiguration of homomorphisms of digraphs. Inspired by notions of directed graph colorings we study the connectivity of Hom⃗(G,Tn) for Tn a tournament, obtaining a complete answer for the case of transitive Tn. If G=Tm is also a transitive tournament, we describe a connection to mixed subdivisions of dilated simplices. Finally we use paths in the internal hom objects of digraphs to define various notions of homotopy, and discuss connections to the topology of Hom⃗ complexes.

Probabilistic Model for rGO-enriched nano-ceramics for electro-mechanical study

Reduced graphene oxide (rGO) nano-ceramics are an extensive area of research because of their fascinating thermal, mechanical, and electrical characteristics. It is possible to undertake a thorough theoretical investigation of the structure's, structural entropy and electrical conductivity by expressing the rGO-included ceramic structure as a polyhedral complex. The work illustrates a theoretical method for choosing the best rGO proportion in order to maximise structural and electrical attributes.

Integration of vector fields on cell complexes and Morse theory

In this paper, we investigate vector fields on polyhedral complexes and their associated trajectories. We study vector fields which are analogues of the gradient vector field of a function in the smooth case. Our goal is to define a nice theory of trajectories of such vector fields, so that the set of them captures the topology of the polyhedral complex, as in classical Morse theory. Since we do not assume the polyhedral complex to be a manifold, the definition of vector fields on it is very different from the smooth case. Nevertheless, we will show that there exist nice classes of functions and metrics which give gradient vector fields with desired properties. Our construction relies on Forman's discrete Morse theory. In particular, the class of functions we use is an improvement of Forman's discrete Morse functions. A notable feature of our theory is that our gradient vector fields are defined purely from functions and metrics as in the smooth case, contrary to the case of discrete Morse theory where we need the data of dimension of cells. This allows us to implement several useful constructions which were not available in the discrete case.

On the Activation Space of ReLU equipped Deep Neural Networks

Modern Deep Neural Networks are getting wider and deeper in their architecture design. However, with an increasing number of parameters the decision mechanisms becomes more opaque. Therefore, there is a need for understanding the structures arising in the hidden layers of deep neural networks.In this work, we present a new mathematical framework for describing the canonical polyhedral decomposition in the input space, and in addition, we introduce the notions of collapsing- and preserving patches, pertinent to understanding the forward map and the activation space they induce.The activation space can be seen as the output of a layer and, in the particular case of ReLU activations, we prove that this output has the structure of a polyhedral complex.

The fiber of persistent homology for simplicial complexes

We study the inverse problem for persistent homology: For a fixed simplicial complex K , we analyze the fiber of the continuous map PH on the space of filters that assigns to a filter f : K → R the total barcode of its associated sublevel set filtration of K . We find that PH is best understood as a map of stratified spaces. Over each stratum of the barcode space the map PH restricts to a (trivial) fiber bundle with fiber a polyhedral complex. Amongst other we derive a bound for the dimension of the fiber depending on the number of distinct endpoints in the barcode. Furthermore, taking the inverse image PH − 1 can be extended to a monodromy functor on the (entrance path) category of barcodes. We demonstrate our theory on the example of the simplicial triangle giving a complete description of all fibers and monodromy maps. This example is rich enough to have a Möbius band as one of its fibers.

The polyhedral tree complex

The tree complex is a simplicial complex defined in recent work of Belk, Lanier, Margalit, and Winarski with applications to mapping class groups and complex dynamics. This article introduces a connection between this setting and the convex polytopes known as associahedra and cyclohedra. Specifically, we describe a characterization of these polytopes using planar embeddings of trees and show that the tree complex is the barycentric subdivision of a polyhedral cell complex for which the cells are products of associahedra and cyclohedra.Mathematics Subject Classifications: 05C05, 05C10, 20F65, 52B11Keywords: Associahedra, cyclohedra, planar trees, mapping class groups

Varieties of tropical ideals are balanced

Tropical ideals, introduced in [12], define subschemes of tropical toric varieties. We prove that the top-dimensional parts of their varieties are balanced polyhedral complexes of the same dimension as the ideal. This means that every subscheme of a tropical toric variety defined by a tropical ideal has an associated class in the Chow ring of the toric variety. A key tool in the proof is that specialization of variables in a tropical ideal yields another tropical ideal; this plays the role of hyperplane sections in the theory. We also show that elimination theory (projection of varieties) works for tropical ideals as in the classical case. The matroid condition that defines tropical ideals is crucial for these results.

Smooth tropical complete intersection curves of genus 3 in $$\mathbb {R}^3$$

We develop a method for describing the tropical complete intersection of a tropical hypersurface and a tropical plane in \(\mathbb {R}^3\). This involves a method for determining the topological type of the intersection of a tropical plane curve and \(\mathbb {R}_{\le 0}^2\) by using a polyhedral complex. As an application, we study smooth tropical complete intersection curves of genus 3 in \(\mathbb {R}^3\). In particular, we show that there are no smooth tropical complete intersection curves in \(\mathbb {R}^3\) whose skeletons are the lollipop graph of genus 3. This gives a partial answer to a problem of Morrison in [6].

Algebraic methods to study the dimension of supersmooth spline spaces

Multivariate piecewise polynomial functions (or splines) on polyhedral complexes have been extensively studied over the past decades and find applications in diverse areas of applied mathematics including numerical analysis, approximation theory, and computer aided geometric design. In this paper we address various challenges arising in the study of splines with enhanced mixed (super-)smoothness conditions at the vertices and across interior faces of the partition. Such supersmoothness can be imposed but can also appear unexpectedly on certain splines depending on the geometry of the underlying polyhedral partition. Using algebraic tools, a generalization of the Billera–Schenck–Stillman complex that includes the effect of additional smoothness constraints leads to a construction which requires the analysis of ideals generated by products of powers of linear forms in several variables. Specializing to the case of planar triangulations, a combinatorial lower bound on the dimension of splines with supersmoothness at the vertices is presented, and we also show that this lower bound gives the exact dimension in high degree. The methods are further illustrated with several examples.

Normal cones corresponding to credal sets of lower probabilities

Credal sets are one of the most important models for describing probabilistic uncertainty. They usually arise as convex sets of probabilistic models compatible with judgments provided in terms of coherent lower previsions or more specific models such as coherent lower probabilities or probability intervals. In finite spaces, credal sets usually take the form of convex polytopes. Many properties of convex polytopes can be derived from their normal cones, which form polyhedral complexes called normal fans. We analyze the properties of normal cones corresponding to credal sets of coherent lower probabilities. For two important classes of coherent lower probabilities, 2-monotone lower probabilities and probability intervals, we provide a detailed description of the normal fan structure. These structures are related to the structure of the extreme points of the credal sets. To arrive at our main results, we provide some general results on triangulated normal fans of convex polyhedra and their adjacency structure.

A New Face Iterator for Polyhedra and for More General Finite Locally Branched Lattices

We discuss a new memory-efficient depth-first algorithm and its implementation that iterates over all elements of a finite locally branched lattice. This algorithm can be applied to face lattices of polyhedra and to various generalizations such as finite polyhedral complexes and subdivisions of manifolds, extended tight spans and closed sets of matroids. Its practical implementation is very fast compared to state-of-the-art implementations of previously considered algorithms. Based on recent work of Bruns, García-Sánchez, O’Neill, and Wilburne, we apply this algorithm to prove Wilf’s conjecture for all numerical semigroups of multiplicity 19 by iterating through the faces of the Kunz cone and identifying the possible bad faces and then checking that these do not yield counterexamples to Wilf’s conjecture.