Discrete homology computations by reduction to zero differentials

2026/08/04 by Sterling Ebel, Chris Kapulkin, Nathan Kershaw
Computer Science · Mathematics · #cs.CG #math.AT #math.CO #msc:05-04 #msc:55N31 #msc:68U05 #msc:05C25 #msc:62R40 #msc:68R05

paper · pdf

21 pages; comments welcome

arxiv created 2026/08/04 · arxiv updated 2026/08/06

Abstract

We develop a new algorithm for computing (persistent) discrete homology of graphs using reduction to zero differentials and active enumeration. This allows us to compute the fourth homology group of the Greene sphere, along with several previously unknown groups. We also show that persistent discrete homology computes faster than simplicial homology of Vietoris-Rips complex in the high-noise non-metric settings, making it a better choice for noisy data sets.

Citations

• Fast computation of the first discrete homology group
• On the homology of simplicial and cubical sets with symmetries
• Topological data analysis using persistent discrete homology
• Keeping it sparse: Computing Persistent Homology revisited
• Keeping it sparse: Computing Persistent Homology revisited
• Cubical setting for discrete homotopy theory, revisited
• An efficient algorithm for 1-dimensional (persistent) path homology
• On the vanishing of discrete singular cubical homology for graphs
• Ripser: efficient computation of Vietoris-Rips persistence barcodes
• Digital Hurewicz theorem and digital homology theory
• Homology groups of cubical sets with connections
• Discrete Cubical and Path Homologies of Graphs
• Persistent Path Homology of Directed Networks
• A roadmap for the computation of persistent homology
• Homotopy theory for digraphs
• Discrete Homology Theory for Metric Spaces
• Computing Topological Persistence for Simplicial Maps
• Dualities in persistent (co)homology
• k-Parabolic Subspace Arrangements
• Discrete Morse Theory for free chain complexes
• A Homotopy Theory for Graphs
• Three-dimensional alpha shapes
• Three-dimensional alpha shapes