The depth poset under transpositions in the filter LIPIcs Edelsbrunner, Herbert ; https://orcid.org/0000-0002-9823-6833 Lipiński, Michał ; https://orcid.org/0000-0001-9789-9750 Mrozek, Marian ; https://orcid.org/0000-0002-0619-6417 Soriano Trigueros, Manuel ; https://orcid.org/0000-0003-2449-1433 Zimin, Fedor Algebraic topology Lefschetz complexes persistent homology vines and vineyards birth-death pairs shallow pairs relations partial orders transpositions Theory of computation → Computational geometry ddc:500 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik 2026 info:eu-repo/semantics/conferenceObject doc-type:conferenceObject text http://purl.org/coar/resource_type/c_5794 https://research-explorer.ista.ac.at/record/22299 https://research-explorer.ista.ac.at/download/22299/22329 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> eng info:eu-repo/semantics/altIdentifier/doi/10.4230/LIPICS.SOCG.2026.41 info:eu-repo/semantics/altIdentifier/e-issn/1868-8969 info:eu-repo/semantics/altIdentifier/isbn/9783959774185 info:eu-repo/semantics/altIdentifier/arxiv/2511.21961 https://creativecommons.org/licenses/by/4.0/ info:eu-repo/semantics/openAccess