NDLI: Nerve Complexes of Circular Arcs

Content Provider	Springer Nature Link
Author	Adams, Henry Peterson, Chris Frick, Florian Adamaszek, Michał Previte Johnson, Corrine
Copyright Year	2016
Abstract	We show that the nerve and clique complexes of n arcs in the circle are homotopy equivalent to either a point, an odd-dimensional sphere, or a wedge sum of spheres of the same even dimension. Moreover this homotopy type can be computed in time $$O(n\log n)$$ . For the particular case of the nerve complex of evenly-spaced arcs of the same length, we determine explicit homology bases and we relate the complex to a cyclic polytope with n vertices. We give three applications of our knowledge of the homotopy types of nerve complexes of circular arcs. First, we show that the Lovász bound on the chromatic number of a circular complete graph is either sharp or off by one. Second, we use the connection to cyclic polytopes to give a novel topological proof of a known upper bound on the distance between successive roots of a homogeneous trigonometric polynomial. Third, we show that the Vietoris–Rips or ambient Čech simplicial complex of n points in the circle is homotopy equivalent to either a point, an odd-dimensional sphere, or a wedge sum of spheres of the same even dimension, and furthermore this homotopy type can be computed in time $$O(n\log n)$$ .
Ending Page	273
Page Count	23
Starting Page	251
File Format	PDF
ISSN	01795376
e-ISSN	14320444
Journal	Discrete & Computational Geometry
Issue Number	2
Volume Number	56
Language	English
Publisher	Springer US
Publisher Date	2016-07-13
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Cyclic polytope Vietoris–Rips complex Computational Mathematics and Numerical Analysis Symmetry properties of polytopes Nerve complex Čech complex Combinatorial aspects of simplicial complexes Circular arc Combinatorics
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Theoretical Computer Science Computational Theory and Mathematics Geometry and Topology

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1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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Vietoris–Rips Complexes of Planar Point Sets

Extremal Betti Numbers of Vietoris–Rips Complexes

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Nerve Complexes of Circular Arcs

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