The Vietoris-Rips Complex of the Circle
Consider a Vietoris-Rips complex of the circle with the geodesic metric. This simplicial complex has an infinite number of vertices, one for each point in the circle. A theorem of Jean-Claude Hausmann implies that for small connectivity parameter, the Vietoris-Rips complex is homotopy equivalent to a circle. What happens as the connectivity parameter increases? We show that the Vietoris-Rips complex obtains the homotopy type of the circle, the 3-sphere, the 5-sphere, the 7-sphere, …, until finally it is contractible.
In particular, we describe the persistent homology of the Vietoris-Rips complex of the circle of unit circumference. The persistence diagram for (2k+1)-dimensional homology consists of the open interval with birth time k/(2k+1) and death time (k+1)/(2k+3).
Joint work-in-progress with Michal Adamaszek, Christopher Peterson, and Corrine Previte.