Related papers: Homological Coordinatization
Various simplicial complexes can be associated with a graph. Box complexes form an important families of such simplicial complexes and are especially useful for providing lower bounds on the chromatic number of the graph via some of their…
This paper lays the foundations of an approach to applying Gromov's ideas on quantitative topology to topological data analysis. We introduce the "contiguity complex", a simplicial complex of maps between simplicial complexes defined in…
We define a simpler notion of symmetric topological complexity more ad hoc to the motion planning problem which was the original motivation for the definition of topological complexity. This is a homotopy invariant that we call…
Node-link diagrams are a popular method for representing graphs that capture relationships between individuals, businesses, proteins, and telecommunication endpoints. However, node-link diagrams may fail to convey insights regarding graph…
While the majority of approaches to the characterization of complex networks has relied on measurements considering only the immediate neighborhood of each network node, valuable information about the network topological properties can be…
Attempts to incorporate topological information in supervised learning tasks have resulted in the creation of several techniques for vectorizing persistent homology barcodes. In this paper, we study thirteen such methods. Besides describing…
We give a framework for constructing generically optimal homotopies for parametrized polynomial systems from tropical data. Here, generically optimal means that the number of paths tracked is equal to the generic number of solutions. We…
Simplicial complexes are higher-order combinatorial structures which have been used to represent real-world complex systems. In this paper, we concentrate on the local patterns in simplicial complexes called simplets, a generalization of…
In this paper, we introduce and study sequential versions of several fibrewise homotopy invariants, including parametrized topological complexity, parametrized (subspace) homotopic distance. We investigate their basic properties, establish…
We study the homology of simplicial and cubical sets with symmetries. These are simplicial and cubical sets with additional maps expressing the symmetries of simplices and cubes. We consider the chain complex computing the homology groups…
We present an algorithm to compute path homology for simple digraphs, and use it to topologically analyze various small digraphs en route to an analysis of complex temporal networks which exhibit such digraphs as underlying motifs. The…
We introduce the weighted path homology on the category of weigh\-ted directed hypergraphs and describe conditions of homotopy invariance of weighted path homology groups. We give several examples that explain the nontriviality of the…
The purpose of this paper is to introduce a version of singular homology based on smooth mappings of manifolds with corners. Although variants of such a theory exists in the literature, we felt that certain points were not adequately…
The aim of this paper is to present a simple way to generate proper monomial rational maps between generalized balls and via the relations between generalized balls and bounded symmetric domains of type I, we suggest new examples of proper…
The purpose of this note is to point out that simplicial methods and the well-known Dold-Kan construction in simplicial homotopy theory can be fruitfully applied to convert link homology theories into homotopy theories. Dold and Kan prove…
A new method is given for computing generators of the homology groups with integer coefficients for any finite $T_0$-space. An important role in this method is played by irreducible cycles which are defined here and give rise to continuous…
Simplicial complexes capture the underlying network topology and geometry of complex systems ranging from the brain to social networks. Here we show that algebraic topology is a fundamental tool to capture the higher-order dynamics of…
We describe various path homology theories constructed for a directed hypergraph. We introduce the category of directed hypergraphs and the notion of a homotopy in this category. Also, we investigate the functoriality and the homotopy…
The classifying spaces of cobordisms of singular maps have two fairly different constructions. We expose a homotopy theoretical connection between them. As a corollary we show that the classifying spaces in some cases have a simple product…
Let F denote the homotopy fiber of a map f:K-->L of 2-reduced simplicial sets. Using as input data the strongly homotopy coalgebra structure of the chain complexes of K and L, we construct a small, explicit chain algebra, the homology of…