2024/06/03 by Vinay Sipani, Sipani, Vinay, Ramesh Kasilingam +1
Biochemistry, Genetics and Molecular Biology · Chemistry · Medicine · #05C10 #05C12 #05E45 #57M15 #68U05 #Algebraic Topology (math.AT) #Click Chemistry and Applications #Combinatorics (math.CO) #DNA and Nucleic Acid Chemistry #FOS: Mathematics #Geometric Topology (math.GT) #Metal complexes synthesis and properties #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2406.01082
openalex publication_date 2024/06/03 · openalex created_date 2024/06/07 · openalex updated_date 2026/07/28
Given a metric space (X,d), the Vietoris-Rips complex of X at a scale of r >0 is a simplicial complex whose simplices are all those finite subsets of X with diameter less than r. In this paper, we classify, up to simplicial isomorphism, all n-dimensional pseudomanifolds and weak-pseudomanifolds that can be realized as a Vietoris-Rips complex of planar point sets. We further classify two-dimensional, pure, and closed planar-Rips complexes up to homotopy. Additionally, we explore the hereditary properties and introduce the notion of obstructions in planar-Rips complexes. We also consolidate our findings to describe a class of unit disk graphs, having all maximal cliques of same cardinality. Several structural and geometric properties of planar-Rips complexes have also been derived.