Does anyone know if the barcode decomposition of a simplex-wise filtration a multiset? More specifically, can we have multiple barcodes with the same birth time? When I read the paper by Gunnar Carlsson: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/BB0DA0F0EBD79809C563AF80B555A23C/S0962492914000051a.pdf/topological_pattern_recognition_for_point_cloud_data.pdf, no restriction was placed on the filtration of simplicial complexes. However, there are other papers that impose a simplex-wise restriction such as the following: https://arxiv.org/pdf/1712.05103.pdf. How does this restriction affect the barcode decomposition of a persistent module?
2026-03-25 15:46:17.1774453577
Barcodes Decomposition of Persistent Homology
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Barcodes are multiset. See here. For example consider 4 points on the vertices of a rectangle. The one dimensional homology class will be born on the shorter edge pair at the same time and die at the same time. Thus you will have two barcodes with same birth and death time.