[{"status":"public","acknowledgement":"The authors thank Jakub Leśkiewicz and Bartosz Furmanek for discussions\r\nthat helped improve the paper. Herbert Edelsbrunner: DFG Collaborative Research Center TRR 109, Austrian Science\r\nFund (FWF), grant no. I 02979-N35\r\nMichał Lipiński: European Union’s Horizon 2020 research and innovation programme under the\r\nMarie Skłodowska-Curie Grant Agreement No. 101034413\r\nMarian Mrozek: Polish National Science Center under Opus Grant 2019/35/B/ST1/00874 and Opus\r\nGrant 2025/57/B/ST1/00550","article_processing_charge":"Yes","corr_author":"1","date_created":"2026-07-13T09:56:38Z","license":"https://creativecommons.org/licenses/by/4.0/","oa":1,"day":"27","publisher":"Schloss Dagstuhl – Leibniz-Zentrum für Informatik","_id":"22299","abstract":[{"lang":"eng","text":"The depth poset of a filtered Lefschetz complex reflects the dependencies between the cancellations of different shallow birth-death pairs. Using the fast algorithms for computing the depth poset in [Edelsbrunner et al., 2026] and for updating the persistence diagram under transpositions in [Cohen-Steiner et al., 2006], we give a complete case analysis of how transpositions of cells in the filter affect the depth poset. In addition, we present statistics on the depth poset for random point data and its sensitivity to the transpositions that occur in random straight-line homotopies."}],"date_updated":"2026-07-14T06:09:32Z","publication_identifier":{"eissn":["1868-8969"],"isbn":["9783959774185"]},"language":[{"iso":"eng"}],"volume":367,"citation":{"short":"H. Edelsbrunner, M. Lipiński, M. Mrozek, M. Soriano Trigueros, F. Zimin, in:, 42nd International Symposium on Computational Geometry, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2026.","mla":"Edelsbrunner, Herbert, et al. “The Depth Poset under Transpositions in the Filter.” <i>42nd International Symposium on Computational Geometry</i>, vol. 367, 41:1-41:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2026, doi:<a href=\"https://doi.org/10.4230/LIPICS.SOCG.2026.41\">10.4230/LIPICS.SOCG.2026.41</a>.","ieee":"H. Edelsbrunner, M. Lipiński, M. Mrozek, M. Soriano Trigueros, and F. Zimin, “The depth poset under transpositions in the filter,” in <i>42nd International Symposium on Computational Geometry</i>, New Brunswick, NJ, United States, 2026, vol. 367.","ista":"Edelsbrunner H, Lipiński M, Mrozek M, Soriano Trigueros M, Zimin F. 2026. The depth poset under transpositions in the filter. 42nd International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 367, 41:1-41:18.","ama":"Edelsbrunner H, Lipiński M, Mrozek M, Soriano Trigueros M, Zimin F. The depth poset under transpositions in the filter. In: <i>42nd International Symposium on Computational Geometry</i>. Vol 367. Schloss Dagstuhl – Leibniz-Zentrum für Informatik; 2026. doi:<a href=\"https://doi.org/10.4230/LIPICS.SOCG.2026.41\">10.4230/LIPICS.SOCG.2026.41</a>","chicago":"Edelsbrunner, Herbert, Michał Lipiński, Marian Mrozek, Manuel Soriano Trigueros, and Fedor Zimin. “The Depth Poset under Transpositions in the Filter.” In <i>42nd International Symposium on Computational Geometry</i>, Vol. 367. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2026. <a href=\"https://doi.org/10.4230/LIPICS.SOCG.2026.41\">https://doi.org/10.4230/LIPICS.SOCG.2026.41</a>.","apa":"Edelsbrunner, H., Lipiński, M., Mrozek, M., Soriano Trigueros, M., &#38; Zimin, F. (2026). The depth poset under transpositions in the filter. In <i>42nd International Symposium on Computational Geometry</i> (Vol. 367). New Brunswick, NJ, United States: Schloss Dagstuhl – Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPICS.SOCG.2026.41\">https://doi.org/10.4230/LIPICS.SOCG.2026.41</a>"},"file_date_updated":"2026-07-14T06:08:05Z","doi":"10.4230/LIPICS.SOCG.2026.41","OA_type":"gold","title":"The depth poset under transpositions in the filter","has_accepted_license":"1","ddc":["500"],"OA_place":"publisher","publication":"42nd International Symposium on Computational Geometry","ec_funded":1,"date_published":"2026-05-27T00:00:00Z","das_tickbox":"0","type":"conference","researchdata_availability":"no","scopus_import":"1","file":[{"checksum":"9dfb96ee66985c724b499b0e5888dc8e","file_name":"2026_LIPIcSSoCG_Edelsbrunner.pdf","date_created":"2026-07-14T06:08:05Z","file_id":"22329","access_level":"open_access","creator":"dernst","success":1,"file_size":2902144,"relation":"main_file","date_updated":"2026-07-14T06:08:05Z","content_type":"application/pdf"}],"oa_version":"Published Version","supplementarymaterial":"no","publication_status":"published","intvolume":"       367","project":[{"grant_number":"I02979-N35","call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"},{"call_identifier":"H2020","grant_number":"101034413","name":"IST-BRIDGE: International postdoctoral program","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c"}],"external_id":{"arxiv":["2511.21961"]},"author":[{"last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert"},{"last_name":"Lipiński","orcid":"0000-0001-9789-9750","id":"dfffb474-4317-11ee-8f5c-fe3fc95a425e","first_name":"Michał","full_name":"Lipiński, Michał"},{"last_name":"Mrozek","orcid":"0000-0002-0619-6417","full_name":"Mrozek, Marian","first_name":"Marian"},{"last_name":"Soriano Trigueros","orcid":"0000-0003-2449-1433","first_name":"Manuel","id":"15ebd7cf-15bf-11ee-aebd-bb4bb5121ea8","full_name":"Soriano Trigueros, Manuel"},{"last_name":"Zimin","id":"afd27eda-91c1-11f0-aad8-c6edbec24c04","first_name":"Fedor","full_name":"Zimin, Fedor"}],"arxiv":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","keyword":["Algebraic topology","Lefschetz complexes","persistent homology","vines and vineyards","birth-death pairs","shallow pairs","relations","partial orders","transpositions","Theory of computation → Computational geometry"],"month":"05","year":"2026","quality_controlled":"1","tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"conference":{"name":"SoCG: Symposium on Computational Geometry","location":"New Brunswick, NJ, United States","start_date":"2026-06-02","end_date":"2026-06-05"},"article_number":"41:1-41:18","department":[{"_id":"HeEd"},{"_id":"GradSch"}],"alternative_title":["LIPIcs"]}]