Discrete Morse Functions Research Articles

The discrete flow category: structure and computation

In this article, we use concepts and methods from the theory of simplicial sets to study discrete Morse theory. We focus on the discrete flow category introduced by Vidit Nanda, and investigate its properties in the case where it is defined from a discrete Morse function on a regular CW complex. We design an algorithm to efficiently compute the Hom posets of the discrete flow category in this case. Furthermore, we show that in the special case where the discrete Morse function is defined on a simplicial complex, then each Hom poset has the structure of a face poset of a regular CW complex. Finally, we prove that the spectral sequence associated to the double nerve of the discrete flow category collapses on page 2.

Discrete Morse theory segmentation on high-resolution 3D lithic artifacts

Abstract Motivated by the question of understanding the roots of tool making by anatomically modern humans and coexisting Neanderthals in the Paleolithic, a number of shape classification methods have been tested on photographs and drawings of stone tools. Since drawings contain interpretation and photographs fool both human and computational methods by color and shadows on the surface, we propose an approach using 3D datasets as best means for analyzing shape, and rely on first open access repositories on lithic tools. The goal is to not only analyze shape on an artifact level, but allow a more detailed analysis of stone tools on a scar and ridge level. A Morse-Smale complex (MS complex) extracted from the triangular mesh of a 3D model is a reduced skeleton consisting of linked lines on the mesh. Discrete Morse theory makes it possible to obtain such a MS complex from a scalar function. Thus, we begin with Multi-Scale Integral Invariant filtering on the meshes of lithic artifacts, which provides curvature measures for ridges, which are convex, and scars, which are concave. The resulting values on the vertices can be used as our discrete Morse function and the skeleton we get is build up from lines that will coincide with the ridges and, implicitly, contains the scars as enclosed regions of those lines on the mesh. As this requires a few parameters, we provide a graphical user interface (GUI) to allow altering the predefined parameters to quickly find a good result. In addition, a stone tool may have areas that do not belong to the scar/ridge class. These can be masked and we use conforming MS complexes to ensure that the skeleton keeps these areas whole. Finally, results are shown on real and open access datasets. The source code and manually annotated ground truth for the evaluation are provided as Open Access with a Creative Commons license.

Persistent pairs and connectedness in discrete Morse functions on simplicial complex I

In this paper, we study some useful properties of persistent pairs in a discrete Morse function on a simplicial complex K. In case of dim⁡K=1 (i.e., a graph), by using the properties, we characterize strongly connectedness of critical simplices between two distinct discrete Morse functions, and relate the number of such pairs to the Euler characteristic of K.

Discrete Morse Functions and Watersheds

Discrete Morse Functions and Watersheds

Witten Complex of Transitive Digraph and Its Convergence

Digraphs are generalization of graphs in which each edge is given one or two directions. For each digraph, there exists a transitive digraph containing it. Moreover, all the formal linear combinations of allowed elementary paths form a basis of the path complex for a transitive digraph. Hence, the study of discrete Morse theory on transitive digraphs is very important for the further study of discrete Morse theory on general digraphs. As we know, the definition of discrete Morse function on a digraph is different from that on a simplical complex or a cell complex: each discrete Morse function on a digraph is a discrete flat Witten-Morse function. In this paper, we deform the usual boundary operator, replacing it with a boundary operator with parameters and consider the induced Laplace operators. In addition, we consider the eigenvectors of the eigenvalues of the Laplace operator that approach to zero when the parameters approach infinity, define the generation space of these eigenvectors, and further give the Witten complex of digraphs. Finally, we prove that for a transitive digraph, Witten complex approaches to its Morse complex. However, for general digraphs, the structure of Morse complex is not as simple as that of transitive digraphs and the critical path is not directly related to the eigenvector with zero eigenvalue of Laplace operator. This is explained in the last part of the paper.

Study on Intelligent Analysis and Processing Technology of Computer Big Data Based on Clustering Algorithm

Aim: Clustering belongs to unsupervised learning, which divides the data objects into the data set into multiple clusters or classes, so that the objects in the same cluster have high similarity Background: The clustering of spatial data objects can be solved by optimization based on the clustering objective function. Objective: Study on intelligent analysis and processing technology of computer big data based on clustering algorithm. Methods: First, a new dynamic self-organizing feature mapping model is proposed, and the training algorithm of the model is given. Then, the spectral clustering technology and related concepts are introduced. The spectral clustering algorithm is studied and analyzed, and a spectral clustering algorithm that automatically determines the number of clusters is proposed. Furthermore, an algorithm for constructing a discrete Morse function to find the optimal solution is proposed, proving that the constructed function is the optimal discrete Morse function. At the same time, two optimization models based on the discrete Morse theory are constructed. Finally, the optimization model based on discrete Morse theory is applied to cluster analysis, and a density clustering algorithm based on the discrete Morse optimization model is proposed. Results: This study is focused on designing and implementing a partitional-based clustering algorithm based on big data, that is suitable for clustering huge datasets to meet low computational requirements. The experiments are conducted in terms of time and space complexity and it is observed that the measure of clustering quality and the run time is capable of running in very less time without negotiating the quality of clustering. The results show that the experiments are carried out on the artificial data set and the UCI data set. Conclusion: The efficiency and superiority of the new model, are verified by comparing it with the clustering results of the DBSCAN algorithm.

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.

Computing Persistent Homology by Spanning Trees and Critical Simplices.

Topological data analysis can extract effective information from higher-dimensional data. Its mathematical basis is persistent homology. The persistent homology can calculate topological features at different spatiotemporal scales of the dataset, that is, establishing the integrated taxonomic relation among points, lines, and simplices. Here, the simplicial network composed of all-order simplices in a simplicial complex is essential. Because the sequence of nested simplicial subnetworks can be regarded as a discrete Morse function from the simplicial network to real values, a method based on the concept of critical simplices can be developed by searching all-order spanning trees. Employing this new method, not only the Morse function values with the theoretical minimum number of critical simplices can be obtained, but also the Betti numbers and composition of all-order cavities in the simplicial network can be calculated quickly. Finally, this method is used to analyze some examples and compared with other methods, showing its effectiveness and feasibility.

On merge trees and discrete Morse functions on paths and trees

In this work we answer an open question asked by Johnson–Scoville. We show that each merge tree is represented by a discrete Morse function on a path. Furthermore, we present explicit constructions for two different but related kinds of discrete Morse functions on paths that induce any given merge tree. A refinement of the used methods allows us to define notions of equivalence of discrete Morse functions on trees which give rise to a bijection between equivalence classes of discrete Morse functions and isomorphism classes of certain labeled merge trees. We also compare our results to similar ones from the literature, in particular to work by Curry.

Discrete Morse Theory on Join of Digraphs

For given two digraphs, we can construct a larger digraph through join. The two digraphs that make up the join are called the factors of the join. In this paper, we give a necessary and sufficient condition that the function on the join determined by the discrete Morse functions on factors is a discrete Morse function. Moreover, we further prove the discrete Morse theory on join when the factors satisfy certain conditions.

Merge trees in discrete Morse theory

In this paper, we study merge trees induced by a discrete Morse function on a tree. Given a discrete Morse function, we provide a method to constructing an induced merge tree and define a new notion of equivalence of discrete Morse functions based on the induced merge tree. We then relate the matching number of a tree to a certain invariant of the induced merge tree. Finally, we count the number of merge trees that can be induced on a star graph and characterize the induced merge tree.

Frontiers of sphere recognition in practice

Sphere recognition is known to be undecidable in dimensions five and beyond, and no polynomial time method is known in dimensions three and four. Here we report on positive and negative computational results with the goal to explore the limits of sphere recognition from a practical point of view. An important ingredient are randomly constructed discrete Morse functions.

On the topology of bi-cyclopermutohedra

Motivated by the work of Panina and her coauthors on cyclopermutohedron we study a poset whose elements correspond to equivalence classes of partitions of the set $$\{1,\dots , n+1\}$$ up to cyclic permutations and orientation reversion. This poset is the face poset of a regular CW complex which we call bi-cyclopermutohedron and denote it by $$\mathrm {QP}_{n+1}$$ . The complex $$\mathrm {QP}_{n+1}$$ contains subcomplexes homeomorphic to moduli space of certain planar polygons with $$n+1$$ sides up to isometries. In this article we find an optimal discrete Morse function on $$\mathrm {QP}_{n+1}$$ and use it to compute its homology with $${\mathbb {Z}}$$ as well as $${\mathbb {Z}}_2$$ coefficients.

On the homotopy and strong homotopy type of complexes of discrete Morse functions

AbstractIn this paper, we determine the homotopy types of the Morse complexes of certain collections of simplicial complexes by studying dominating vertices or strong collapses. We show that if K contains two leaves that share a common vertex, then its Morse complex is strongly collapsible and hence has the homotopy type of a point. We also show that the pure Morse complex of a tree is strongly collapsible, thereby recovering as a corollary a result of Ayala et al. (2008, Topology and Its Applications 155, 2084–2089). In addition, we prove that the Morse complex of a disjoint union $K\sqcup L$ is the Morse complex of the join $K*L$ . This result is used to compute the homotopy type of the Morse complex of some families of graphs, including Caterpillar graphs, as well as the automorphism group of a disjoint union for a large collection of disjoint complexes.

A Discrete Morse Perspective on Knot Projections and a Generalised Clock Theorem

We obtain a simple and complete characterisation of which matchings on the Tait graph of a knot diagram induce a discrete Morse function (dMf) on the two sphere, extending a construction due to Cohen. We show these dMfs are in bijection with certain rooted spanning forests in the Tait graph. We use this to count the number of such dMfs with a closed formula involving the graph Laplacian. We then simultaneously generalise Kauffman's Clock Theorem and Kenyon-Propp-Wilson's correspondence in two different directions; we first prove that the image of the correspondence induces a bijection on perfect dMfs, then we show that all perfect matchings, subject to an admissibility condition, are related by a finite sequence of click and clock moves. Finally, we study and compare the matching and discrete Morse complexes associated to the Tait graph, in terms of partial Kauffman states, and provide some computations.

Properties of HYMNs in examples of four-color, five-color, and six-color adinkras

The mathematical concept of a “Banchoff index” associated with discrete Morse functions for oriented triangular meshes has been shown to correspond to the height assignments of nodes in adinkras. In recent work there has been introduced the concept of “Banchoff matrices” leading to HYMNs — height yielding matrix numbers. HYMNs map the shape of an adinkra to a set of eigenvalues derived from Banchoff matrices. In the context of some examples of four-color, minimal five-color, and minimal six-color adinkras, properties of the HYMNs are explored.

Relations in doubly laced crystal graphs via discrete Morse theory

We study the combinatorics of crystal graphs given by highest weight representations of types $A_{n}, B_{n}, C_{n}$, and $D_{n}$, uncovering new relations that exist among crystal operators. Much structure in these graphs has been revealed by local relations given by Stembridge and Sternberg. However, there exist relations among crystal operators that are not implied by Stembridge or Sternberg relations. Viewing crystal graphs as edge colored posets, we use poset topology to study them. Using the lexicographic discrete Morse functions of Babson and Hersh, we relate the Mobius function of a given interval in a crystal poset of simply laced or doubly laced type to the types of relations that can occur among crystal operators within this interval. For a crystal of a highest weight representation of finite classical Cartan type, we show that whenever there exists an interval whose Mobius function is not equal to -1, 0, or 1, there must be a relation among crystal operators within this interval not implied by Stembridge or Sternberg relations. As an example of an application, this yields relations among crystal operators in type $C_{n}$ that were not previously known. Additionally, by studying the structure of Sternberg relations in the doubly laced case, we prove that crystals of highest weight representations of types $B_{2}$ and $C_{2}$ are not lattices.

The Realization Problem for Discrete Morse Functions on Trees

We introduce a new notion of equivalence of discrete Morse functions on graphs called persistence equivalence. Two functions are considered persistence equivalent if and only if they induce the same persistence diagram. We compare this notion of equivalence to other notions of equivalent discrete Morse functions. Then we compute an upper bound for the number of persistence equivalent discrete Morse functions on a fixed graph and show that this upper bound is sharp in the case where our graph is a tree. This is a version of the “realization problem” of the persistence map. We conclude with an example illustrating our construction.

Min-Max Theory for Cell Complexes

In the study of smooth functions on manifolds, min-max theory provides a mechanism for identifying critical values of a function. We introduce a discretized version of this theory associated to a discrete Morse function on a (regular) cell complex. As applications we prove a discrete version of the mountain pass lemma and give an alternate proof of a discrete Lusternik–Schnirelmann theorem.

Discrete Morse functions, vector fields, and homological sequences on trees

We construct a discrete Morse function which induces both a specified gradient vector field and homological sequence on a given tree. After reviewing the basics of discrete Morse theory, we provide an algorithm to construct a discrete Morse function on a tree inducing a desired gradient vector field and homological sequence. We prove that our algorithm is correct, and conclude with an example to illustrate its use.