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DOI: 10.4230/LIPIcs.SoCG.2026.41

The Depth Poset Under Transpositions in the Filter

Authors: Herbert Edelsbrunner, Michał Lipiński, Marian Mrozek, Manuel Soriano-Trigueros, and Fedor Zimin

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract

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.

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Herbert Edelsbrunner, Michał Lipiński, Marian Mrozek, Manuel Soriano-Trigueros, and Fedor Zimin. The Depth Poset Under Transpositions in the Filter. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 41:1-41:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{edelsbrunner_et_al:LIPIcs.SoCG.2026.41,
  author =	{Edelsbrunner, Herbert and Lipi\'{n}ski, Micha{\l} and Mrozek, Marian and Soriano-Trigueros, Manuel and Zimin, Fedor},
  title =	{{The Depth Poset Under Transpositions in the Filter}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{41:1--41:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.41},
  URN =		{urn:nbn:de:0030-drops-258479},
  doi =		{10.4230/LIPIcs.SoCG.2026.41},
  annote =	{Keywords: Algebraic topology, Lefschetz complexes, persistent homology, vines and vineyards, birth-death pairs, shallow pairs, relations, partial orders, transpositions}
}

@InProceedings{edelsbrunner_et_al:LIPIcs.SoCG.2026.41,
  author =	{Edelsbrunner, Herbert and Lipi\'{n}ski, Micha{\l} and Mrozek, Marian and Soriano-Trigueros, Manuel and Zimin, Fedor},
  title =	{{The Depth Poset Under Transpositions in the Filter}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{41:1--41:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.41},
  URN =		{urn:nbn:de:0030-drops-258479},
  doi =		{10.4230/LIPIcs.SoCG.2026.41},
  annote =	{Keywords: Algebraic topology, Lefschetz complexes, persistent homology, vines and vineyards, birth-death pairs, shallow pairs, relations, partial orders, transpositions}
}

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The Depth Poset Under Transpositions in the Filter

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