20.500.12556/DiRROS-20227 Persistent homology with selective Rips complexes detects geodesic circles This paper introduces a method to detect each geometrically significant loop that is a geodesic circle (an isometric embedding of $S^1$) and a bottleneck loop (meaning that each of its perturbations increases the length) in a geodesic space using persistent homology. Under fairly mild conditions, we show that such a loop either terminates a 1- dimensional homology class or gives rise to a 2-dimensional homology class in persistent homology. The main tool in this detection technique are selective Rips complexes, new custom made complexes that function as an appropriate combinatorial lens for persistent homology to detect the above mentioned loops. The main argument is based on a new concept of a local winding number, which turns out to be an invariant of certain homology classes. simple closed geodesic Rips complex persistent homology local winding number true false true Angleški jezik Ni določen Neznano 2024-08-26 11:05:34 2024-08-26 11:05:34 2024-08-27 03:35:32 0000-00-00 00:00:00 2024 0 0 23 str. iss. 6, article no. 170 Vol. 21 Sep. 2024 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2024-09-01 515.1 1660-5446 10.1007/s00009-024-02706-0 204600067 s00009-024-02706-0.pdf s00009-024-02706-0.pdf 1 F03192DEB6B7C3388542F1404265A7AC 0541745b3793b23aada88f7c3cc127f6aff16237e29ab6131b0544778c6966fc c0022ed5-6379-11ef-b487-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=28331 https://link.springer.com/article/10.1007/s00009-024-02706-0 1 981e4b00-6379-11ef-b487-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=28330 Inštitut za matematiko, fiziko in mehaniko 0 0 0