Related papers: Flagifying the Dowker Complex
If $K$ is a simplicial complex on $m$ vertices the flagification of $K$ is the minimal flag complex $K^f$ on the same vertex set that contains $K$. Letting $L$ be the set of vertices, there is a sequence of simplicial inclusions $L\to K\to…
The Fast Fourier Transform (FFT) correlation approach to protein-protein docking can evaluate the energies of billions of docked conformations on a grid if the energy is described in the form of a correlation function. Here, this…
The Rips filtration over a finite metric space and its corresponding persistent homology are prominent methods in TDA to summarise the shape of data. Crucial to their use is the bottleneck stability result. A generalisation of the Rips…
The alpha complex is a subset of the Delaunay triangulation and is often used in computational geometry and topology. One of the main drawbacks of using the alpha complex is that it is non-monotone, in the sense that if ${\cal…
A map $f:X\to Y$ to a simplicial complex $Y$ is called a $Y$-triangular homotopy equivalence if it has a homotopy inverse $g$ and homotopies $h_1:f\circ g\simeq \mathrm{id}_Y$, $h_2:g\circ f\simeq \mathrm{id}_X$ such that for all simplices…
In the theory of persistent homology, a well known duality relates the barcodes of the absolute homology and relative cohomology of a one-parameter simplicial filtration. Motivated by the problem of computing free presentations of the…
We provide a recursive construction of an acyclic matching (also known as a gradient vector field, an equivalent notion to a discrete Morse function) on the independence complex of a graph with a simplicial vertex using given acyclic…
The Topological Signal Processing (TSP) framework has been recently developed to analyze signals defined over simplicial complexes, i.e. topological spaces represented by finite sets of elements that are closed under inclusion of subsets…
The dual complex of a singularity is defined, up-to homotopy, using resolutions of singularities. In many cases, for instance for isolated singularities, we identify and study a "minimal" representative of the homotopy class that is well…
We provide a bottom up construction of torsion generators for weighted homology of a weighted complex over a discrete valuation ring $R=\mathbb{F}[[\pi]]$. This is achieved by starting from a basis for classical homology of the $n$-th…
In this work we investigate from a broad perspective the reduction of degrees of freedom through serendipity techniques for polytopal methods compatible with Hilbert complexes. We first establish an abstract framework that, given two…
We consider the problem of efficiently computing a discrete Morse complex on simplicial complexes of arbitrary dimension and very large size. Based on a common graph-based formalism, we analyze existing data structures for simplicial…
Simplicial complexes constitute the underlying topology of interacting complex systems including among the others brain and social interaction networks. They are generalized network structures that allow to go beyond the framework of…
Recent increase in the availability of warped images projected onto a manifold (e.g., omnidirectional spherical images), coupled with the success of higher-order assignment methods, has sparked an interest in the search for improved…
In this paper we continue the programme of topologically twisting N=2 theories in D=4, focusing on the coupling of vector multiplets to N=2 supergravity. We show that in the minimal case, namely when the special geometry prepotential F(X)…
Simplicial complexes can be viewed as high dimensional generalizations of graphs that explicitly encode multi-way ordered relations between vertices at different resolutions, all at once. This concept is central towards detection of higher…
We use the topology of simplicial complexes to model political structures following [1]. Simplicial complexes are a natural tool to encode interactions in the structures since a simplex can be used to represent a subset of compatible…
A doubling covering $\U$ of a complex $n$-dimensional manifold $Y$ consists of analytic functions $\psi_j:B_1\to Y$, each function being analytically extendable, as a mapping to $Y$, to a four times larger concentric ball $B_4$. Main result…
We present a formula for determining synchronizability in large, randomized and weighted simplicial complexes. This formula leverages eigenratios and costs to assess complete synchronizability under diverse network topologies and intensity…
Many practical applications in topological data analysis arise from data in the form of point clouds, which then yield simplicial complexes. The combinatorial structure of simplicial complexes captures the topological relationships between…