Persistent homology with selective Rips complexes detects geodesic circles

Virk, Žiga (author)

This paper introduces a method to detect each geometrically significant loop that is a geodesic circle (an isometric embedding of $S^1$) and a bottleneck loop (meaning that each of its perturbations increases the length) in a geodesic space using persistent homology. Under fairly mild conditions, we show that such a loop either terminates a 1-dimensional homology class or gives rise to a 2-dimensional homology class in persistent homology. The main tool in this detection technique are selective Rips complexes, new custom made complexes that function as an appropriate combinatorial lens for persistent homology to detect the above mentioned loops. The main argument is based on a new concept of a local winding number, which turns out to be an invariant of certain homology classes.

• Language: English
• Keywords: simple closed geodesic spaces, Rips complex, persistent homology, local winding number
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FRI - Faculty of Computer and Information Science
• Publication status: Published
• Publication version: Version of Record
• Year: 2024
• Number of pages: 23 str.
• Numbering: Vol. 21, iss. 6, art. 170
• PID: 20.500.12556/RUL-160126
• UDC: 515.1
• ISSN on article: 1660-5446
• DOI: 10.1007/s00009-024-02706-0
• COBISS.SI-ID: 204600067
• Publication date in RUL: 21.08.2024

Record is a part of a journal

• Title: Mediterranean journal of mathematics
• Shortened title: Mediterr. j. math.
• Publisher: Springer Nature
• ISSN: 1660-5446
• COBISS.SI-ID: 13561433

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Projects

• Funder: ARRS - Slovenian Research Agency
• Project number: J1-4001
• Name: Izbrani problemi iz uporabne in računske topologije

• Funder: ARRS - Slovenian Research Agency
• Project number: P1-0292
• Name: Topologija in njena uporaba