Abstract. Given a metric space X and a distance threshold r > 0, the Vietoris-Rips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of JeanClaude Hausmann states that if X is a Riemannian manifold and r is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. Little is known about the behavior of Vietoris-Rips complexes for larger values of r, even though these complexes arise naturally in applications using persistent homology. We show that as r increases, the VietorisRips complex of the circle obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, . . . , until finally it is contractible. As our main tool we introduce a directed graph invariant, the winding fraction, which in some sense is dual to the circular chromatic number. Using the winding fraction we classify the homotopy types of the Vietoris-Rips complex of an arbitrary (possibly infinite) subset of the circle, and we study the expected homotopy type of the Vietoris-Rips complex of a uniformly random sample from the circle. Moreover, we show that as the distance parameter increases, the ambientČech complex of the circle (i.e. the nerve complex of the covering of a circle by all arcs of a fixed length) also obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, . . . , until finally it is contractible.
Suppose that ball-shaped sensors wander in a bounded domain. A sensor doesn't know its location but does know when it overlaps a nearby sensor. We say that an evasion path exists in this sensor network if a moving intruder can avoid detection. In Coordinate-free coverage in sensor networks with controlled boundaries via homology, Vin de Silva and Robert Ghrist give a necessary condition, depending only on the time-varying connectivity data of the sensors, for an evasion path to exist. Using zigzag persistent homology, we provide an equivalent condition that moreover can be computed in a streaming fashion. However, no method with time-varying connectivity data as input can give necessary and sufficient conditions for the existence of an evasion path. Indeed, we show that the existence of an evasion path depends not only on the fibrewise homotopy type of the region covered by sensors but also on its embedding in spacetime. For planar sensors that also measure weak rotation and distance information, we provide necessary and sufficient conditions for the existence of an evasion path.arXiv:1308.3536v3 [math.AT]
1-12], the authors apply computational topological tools to the data set of optical patches studied by Lee, Pedersen, and Mumford and find geometric structures for high density subsets. One high density subset is called the primary circle and essentially consists of patches with a line separating a light and a dark region. In this paper, we apply the techniques of Carlsson et al. to range patches. By enlarging to 5 × 5 and 7 × 7 patches, we find core subsets that have the topology of the primary circle, suggesting a stronger connection between optical patches and range patches than was found by Lee, Pedersen, and Mumford.
Given a sample of points X in a metric space M and a scale r > 0, the Vietoris-Rips simplicial complex VR(X; r) is a standard construction to attempt to recover M from X up to homotopy type. A deficiency of this approach is that VR(X; r) is not metrizable if it is not locally finite, and thus does not recover metric information about M . We attempt to remedy this shortcoming by defining a metric space thickening of X, which we call the Vietoris-Rips thickening VR m (X; r), via the theory of optimal transport. When M is a complete Riemannian manifold, or alternatively a compact Hadamard space, we show that the the Vietoris-Rips thickening satisfies Hausmann's theorem (VR m (M ; r) ≃ M for r sufficiently small) with a simpler proof: homotopy equivalence VR m (M ; r) → M is canonically defined as a center of mass map, and its homotopy inverse is the (now continuous) inclusion map M ֒→ VR m (M ; r). Furthermore, we describe the homotopy type of the Vietoris-Rips thickening of the n-sphere at the first positive scale parameter r where the homotopy type changes.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.