Simplicial approximation provides a framework for constructing simplicial complexes that are homotopy equivalent to a given manifold, provided a CW structure is explicitly known. However, its conventional implementation quickly becomes intractable on a computer: barycentric subdivision produces poorly shaped simplices, and the star condition introduces many vertices. To address these limitations, this article develops a subdivision scheme based on spherical Delaunay triangulations, which attains better refinement properties than barycentric subdivisions. Moreover, the star condition is reframed as two independent problems, one geometric and the other combinatorial, respectively tackled in the language of locally equiconnected spaces and the list homomorphism problem, allowing an exponential reduction in the number of vertices. Via a prototype implementation, we obtain simplicial complexes homotopy equivalent to Grassmannians and Stiefel manifolds up to dimension 5.

Temporal graphs are commonly used to represent complex systems and track the evolution of their constituents over time. Visualizing these graphs is crucial as it allows one to quickly identify anomalies, trends, patterns, and other properties that facilitate better decision-making. In this context, selecting an appropriate temporal resolution is essential for constructing and visually analyzing the layout. The choice of resolution is particularly important, especially when dealing with temporally sparse graphs. In such cases, changing the temporal resolution by grouping events (i.e., edges) from consecutive timestamps - a technique known as timeslicing - can aid in the analysis and reveal patterns that might not be discernible otherwise. However, selecting an appropriate temporal resolution is a challenging task. In this paper, we propose ZigzagNetVis, a methodology that suggests temporal resolutions potentially relevant for analyzing a given graph, i.e., resolutions that lead to substantial topological changes in the graph structure. ZigzagNetVis achieves this by leveraging zigzag persistent homology, a well-established technique from Topological Data Analysis (TDA). To improve visual graph analysis, ZigzagNetVis incorporates the colored barcode, a novel timeline-based visualization inspired by persistence barcodes commonly used in TDA. We also contribute with a web-based system prototype that implements suggestion methodology and visualization tools. Finally, we demonstrate the usefulness and effectiveness of ZigzagNetVis through a usage scenario, a user study with 27 participants, and a detailed quantitative evaluation.

Given a sample of an abstract manifold immersed in some Euclidean space, we describe a way to recover the singular homology of the original manifold. It consists in estimating its tangent bundle—seen as subset of another Euclidean space—from a measure theoretical point of view, and in applying measure-based filtrations for persistent homology. The construction we propose is consistent and stable, and does not require the dimension of the manifold.