A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs

article A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs LIPIcs submitted yes Ranita Biswas author 3C2B033E-F248-11E8-B48F-1D18A9856A870000-0002-5372-7890 Sebastiano Cultrera di Montesano author 34D2A09C-F248-11E8-B48F-1D18A9856A870000-0001-6249-0832 Herbert Edelsbrunner author 3FB178DA-F248-11E8-B48F-1D18A9856A870000-0002-9823-6833 Morteza Saghafian author GradSch department HeEd department Alpha Shape Theory Extended project Mathematics, Computer Science project Persistence and stability of geometric complexes project We characterize critical points of 1-dimensional maps paired in persistent homology geometrically and this way get elementary proofs of theorems about the symmetry of persistence diagrams and the variation of such maps. In particular, we identify branching points and endpoints of networks as the sole source of asymmetry and relate the cycle basis in persistent homology with a version of the stable marriage problem. Our analysis provides the foundations of fast algorithms for maintaining collections of interrelated sorted lists together with their persistence diagrams. https://research-explorer.ista.ac.at/download/11660/11661/window 1.pdf application/pdfno Schloss Dagstuhl - Leibniz-Zentrum für Informatik2022 eng LIPIcs https://research-explorer.ista.ac.at/record/15094 https://research-explorer.ista.ac.at/record/13182 Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “A Window to the Persistence of 1D Maps. I: Geometric Characterization of Critical Point Pairs.” LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, n.d. R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs,” LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, LIPIcs (n.d.). Biswas, Ranita, et al. “A Window to the Persistence of 1D Maps. I: Geometric Characterization of Critical Point Pairs.” LIPIcs, Schloss Dagstuhl - Leibniz-Zentrum für Informatik. Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs. LIPIcs. Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M. (n.d.). A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs. LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs. LIPIcs. 116602022-07-27T09:31:15Z2026-08-13T09:41:58Z