---
_id: '11660'
abstract:
- lang: eng
  text: '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. '
acknowledgement: 'This project has received funding from the European Research Council
  (ERC) under the European Union’s Horizon 2020 research and innovation programme,
  grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant
  no. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization
  in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35. '
alternative_title:
- LIPIcs
article_processing_charge: No
author:
- first_name: Ranita
  full_name: Biswas, Ranita
  id: 3C2B033E-F248-11E8-B48F-1D18A9856A87
  last_name: Biswas
  orcid: 0000-0002-5372-7890
- first_name: Sebastiano
  full_name: Cultrera di Montesano, Sebastiano
  id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
  last_name: Cultrera di Montesano
  orcid: 0000-0001-6249-0832
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Morteza
  full_name: Saghafian, Morteza
  last_name: Saghafian
citation:
  ama: '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.
    <i>LIPIcs</i>.'
  apa: 'Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian,
    M. (n.d.). A window to the persistence of 1D maps. I: Geometric characterization
    of critical point pairs. <i>LIPIcs</i>. Schloss Dagstuhl - Leibniz-Zentrum für
    Informatik.'
  chicago: '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.” <i>LIPIcs</i>. Schloss Dagstuhl - Leibniz-Zentrum für
    Informatik, n.d.'
  ieee: '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,” <i>LIPIcs</i>. Schloss Dagstuhl - Leibniz-Zentrum für Informatik.'
  ista: '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.'
  mla: 'Biswas, Ranita, et al. “A Window to the Persistence of 1D Maps. I: Geometric
    Characterization of Critical Point Pairs.” <i>LIPIcs</i>, Schloss Dagstuhl - Leibniz-Zentrum
    für Informatik.'
  short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, LIPIcs
    (n.d.).
corr_author: '1'
date_created: 2022-07-27T09:31:15Z
date_published: 2022-07-25T00:00:00Z
date_updated: 2026-08-13T09:41:58Z
day: '25'
ddc:
- '510'
department:
- _id: GradSch
- _id: HeEd
ec_funded: 1
file:
- access_level: open_access
  checksum: 95903f9d1649e8e437a967b6f2f64730
  content_type: application/pdf
  creator: scultrer
  date_created: 2022-07-27T09:30:30Z
  date_updated: 2022-07-27T09:30:30Z
  file_id: '11661'
  file_name: window 1.pdf
  file_size: 564836
  relation: main_file
file_date_updated: 2022-07-27T09:30:30Z
has_accepted_license: '1'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '07'
oa: 1
oa_version: Submitted Version
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '788183'
  name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: Z00342
  name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: I02979-N35
  name: Persistence and stability of geometric complexes
publication: LIPIcs
publication_status: submitted
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
related_material:
  record:
  - id: '15094'
    relation: dissertation_contains
    status: public
  - id: '13182'
    relation: later_version
    status: public
status: public
title: 'A window to the persistence of 1D maps. I: Geometric characterization of critical
  point pairs'
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2022'
...