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Area of Science:

  • Computational Topology
  • Geometric Data Analysis
  • Persistent Homology

Background:

  • The k-fold cover is a topological construction used in data analysis.
  • Persistent homology is a powerful tool for analyzing the shape of data.
  • Standard methods for persistent homology work well for scale-based filtrations.

Purpose of the Study:

  • To compute persistence diagrams for two types of filtrations of k-fold covers: scale and depth.
  • To develop novel geometric and topological methods for the depth filtration.
  • To establish a connection between Delaunay mosaics and the persistence module of multi-covers.

Main Methods:

  • Definition of k-fold cover for a point set X and radius r.
  • Consideration of two filtrations: scale (fixed k, increasing r) and depth (fixed r, decreasing k).
  • Introduction of a rhomboid tiling and order-k Delaunay mosaics for the depth filtration.
  • Construction of a zigzag module of Delaunay mosaics.

Main Results:

  • Computation of persistence diagrams for both scale and depth filtrations.
  • Demonstration that standard methods suffice for scale filtration.
  • Development of new geometric and topological concepts for depth filtration.
  • Isomorphism between the zigzag module of Delaunay mosaics and the persistence module of multi-covers.

Conclusions:

  • The study provides a comprehensive analysis of k-fold covers using persistent homology.
  • Novel methods are introduced for analyzing depth-based filtrations, expanding the applicability of persistent homology.
  • The connection to Delaunay mosaics offers new insights into the structure of multi-covers.