2024/05/15 by Žiga Virk, Virk, Žiga · 1 citation
Computer Science · #Advanced Algebra and Logic #Advanced Graph Theory Research #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2405.09134
openalex publication_date 2024/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that for each positive integer n, the Rips complexes of the n-dimensional integer lattice in the d1 metric (i.e., the Manhattan metric, also called the natural word metric in the Cayley graph) are contractible at scales above n2(2n-1), with the bounds arising from the Jung's constants. We introduce a new concept of locally dominated vertices in a simplicial complex, upon which our proof strategy is based. This allows us to deduce the contractibility of the Rips complexes from a local geometric condition called local crushing. In the case of the integer lattices in dimension n and a fixed scale r, this condition entails the comparison of finitely many distances to conclude that the corresponding Rips complex is contractible. In particular, we are able to verify that for n=1,2,3, the Rips complex of the n-dimensional integer lattice at scale greater or equal to n is contractible. We conjecture that the same proof strategy can be used to extend this result to all dimensions n