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The Depth Poset Under Transpositions in the Filter

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

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LIPIcs.SoCG.2026.41.pdf
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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Algebraic topology
  • Lefschetz complexes
  • persistent homology
  • vines and vineyards
  • birth-death pairs
  • shallow pairs
  • relations
  • partial orders
  • transpositions

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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.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.SoCG.2026.41

Author Details

Herbert Edelsbrunner
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria
Michał Lipiński
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria
Marian Mrozek
  • Division of Computational Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
Manuel Soriano-Trigueros
  • Universidad de Sevilla, Spain
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria
Fedor Zimin
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria

Funding

  • Edelsbrunner, Herbert: DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35
  • Lipiński, Michał: European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 101034413
  • Mrozek, Marian: Polish National Science Center under Opus Grant 2019/35/B/ST1/00874 and Opus Grant 2025/57/B/ST1/00550

Acknowledgements

The authors thank Jakub Leśkiewicz and Bartosz Furmanek for discussions that helped improve the paper.

References

  1. Dominique Attali, Marc Glisse, Samuel Hornus, Francis Lazarus, and Dmitriy Morozov. Persistence-sensitive simplification of functions on surfaces in linear time. Manuscript, presented at TOPOINVIS, 9:23-24, 2009. URL: https://members.loria.fr/SHornus/simplif/attali-pssfslt.pdf.
  2. Ulrich Bauer. Ripser: efficient computation of Vietoris-Rips persistence barcodes. Journal of Applied and Computational Topology, 5(3):391-423, 2021. URL: https://doi.org/10.1007/s41468-021-00071-5.
  3. Ulrich Bauer, Carsten Lange, and Max Wardetzky. Optimal topological simplification of discrete functions on surfaces. Discrete & Computational Geometry, 47(2):347-377, 2012. URL: https://doi.org/10.1007/s00454-011-9350-z.
  4. Martin P. Bendsøe. Optimization of Structural Topology, Shape, and Material. Springer, Berlin, Germany, 1995. URL: https://doi.org/10.1007/978-3-662-03115-5.
  5. Martin P. Bendsøe and Ole Sigmund. Topology Optimization. Springer, Berlin, Germany, 2nd edition, 2004. URL: https://doi.org/10.1007/978-3-662-05086-6.
  6. David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Stability of persistence diagrams. Discrete & Computational Geometry, 37(1):103-120, 2007. URL: https://doi.org/10.1007/s00454-006-1276-5.
  7. David Cohen-Steiner, Herbert Edelsbrunner, and Dmitriy Morozov. Vines and vineyards by updating persistence in linear time. In Proceedings of the 26th Annual Symposium on Computational Geometry, SCG '06, pages 119-126, New York, NY, USA, 2006. Association for Computing Machinery. URL: https://doi.org/10.1145/1137856.1137877.
  8. Olaf Delgado-Friedrichs, Vanessa Robins, and Adrian Sheppard. Skeletonization and partitioning of digital images using discrete Morse theory. IEEE Trans Pattern Anal Mach Intell, 37(3):654-666, March 2015. URL: https://doi.org/10.1109/TPAMI.2014.2346172.
  9. Hebert Edelsbrunner, Michał Lipiński, Marian Mrozek, and Manuel Soriano-Trigueros. The poset of cancellations induced by gradient dynamics in a filtered Lefschetz complex. arXiv:2311.14364, 2026. URL: https://doi.org/10.48550/arXiv.2311.14364.
  10. Herbert Edelsbrunner and John L Harer. Computational topology: an introduction. American Mathematical Society, Providence, Rhode Island, 2010. URL: https://doi.org/10.1090/mbk/069.
  11. Herbert Edelsbrunner, Dmitriy Morozov, and Valerio Pascucci. Persistence-sensitive simplification functions on 2-manifolds. In Proceedings of the Twenty-Second Annual Symposium on Computational Geometry, SCG '06, pages 127-134, New York, NY, USA, 2006. Association for Computing Machinery. URL: https://doi.org/10.1145/1137856.1137878.
  12. Robin Forman. Combinatorial vector fields and dynamical systems. Mathematische Zeitschrift, 228(4):629-681, 1998. URL: https://doi.org/10.1007/PL00004638.
  13. Robin Forman. Morse theory for cell complexes. Advances in Mathematics, 134(1):90-145, 1998. URL: https://doi.org/10.1006/aima.1997.1650.
  14. Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. Google Scholar
  15. Solomon Lefschetz. Algebraic Topology, volume 27 of Colloquium Publications. American Mathematical Society, Providence, Rhode Island, 1942. Google Scholar
  16. Michał Lipiński, Jacek Kubica, Marian Mrozek, and Thomas Wanner. Conley-Morse-Forman theory for generalized combinatorial multivector fields on finite topological spaces. Journal of Applied and Computational Topology, 7:139-184, 2022. URL: https://doi.org/10.1007/s41468-022-00102-9.
  17. Marian Mrozek. Conley-Morse-Forman theory for combinatorial multivector fields on Lefschetz complexes. Foundations of Computational Mathematics, 17(6):1585-1633, 2017. URL: https://doi.org/10.1007/s10208-016-9330-z.
  18. Marian Mrozek and Thomas Wanner. Connection Matrices in Combinatorial Topological Dynamics. SpringerBriefs in Mathematics. Springer Cham, 2025. URL: https://doi.org/10.1007/978-3-031-87600-4.
  19. Arnur Nigmetov and Dmitriy Morozov. Topological optimization with big steps. Discrete & Computational Geometry, 72(1):310-344, 2024. URL: https://doi.org/10.1007/s00454-023-00613-x.
  20. Jens Kristian Refsgaard Schou and Bei Wang. Persisort: A new perspective on adaptive sorting based on persistence. In Proc. 36th Canadian Conference on Computational Geometry, pages 287-302, July 2024. Google Scholar
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