2025/11/06 by Raju Kumar Gupta, Sourav Sarkar, Gupta, Raju Kumar +3 · 1 citation
Computer Science · Mathematics · #20F65 #51M05 #52C07 #55N31 #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #Combinatorics (math.CO) #Data Management and Algorithms #FOS: Mathematics #Geometric Topology (math.GT) #Metric Geometry (math.MG) #Primary: 55P10 #Secondary: 57M07 #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2511.04238
openalex publication_date 2025/11/06 · openalex created_date 2025/11/08 · openalex updated_date 2026/07/28
For a metric space X and r ≥ 0, the Vietoris-Rips complex VR(X;r) is a simplicial complex whose simplices are finite subsets of X with diameter at most r. Vietoris-Rips complexes have applications in various places, including data analysis, geometric group theory, sensor networks, etc. Consider the integer lattice ℤn as a metric space equipped with the d1-metric (the Manhattan metric or standard word metric in the Cayley graph). Ziga Virk proved that if either r ≥ n2(2n-1), or 1≤ n ≤ 3 and r ≥ n, then the complex VR(ℤn;r) is contractible, and posed a question if VR(ℤn;r) is contractible for all r ≥ n. Recently, Matthew Zaremsky improved Ziga's result and proved that VR(ℤn;r) is contractible if r ≥ n2+ n-1. Further, he conjectured that VR(ℤn;r) is contractible for all r ≥ n. We prove Zaremsky's conjecture for n ≤ 5, i.e., we prove that VR(ℤn;r) is contractible if n ≤ 5 and r ≥ n. Further, we prove that VR(ℤn;r) is contractible for r ≥ 10. We determine the homotopy type of VR(ℤn;2), and show that these complexes are homotopy equivalent to a wedge of countably infinite copies of \mathbbS3. We also show that VR(ℤn;r) is simply connected for r ≥ 2.