What is the Vietoris-Rips Complex for Evenly Spaced Points Around a Circle?
Tuesday, October 1, 2013 - 1:30pm - 2:30pm
Consider the Vietoris-Rips complex for n evenly-spaced points around a circle. For small choices of the connectivity parameter, this complex is homotopy equivalent to a circle. For n even and for a connectivity parameter slightly less than the diameter of the circle, this complex is the boundary of a cross-polytope and is homeomorphic to the (n/2-1)-sphere. What happens in the intermediate range when the connectivity parameter is neither small nor large? We provide evidence for the following claim: for the correct choice of connectivity parameter and for t>1, the Vietoris-Rips complex on (2m+1)t evenly-spaced points around a circle is homotopy equivalent to the (t-1)-fold wedge sum of 2m-spheres. This is joint work-in-progress with Cory Previte, Chris Peterson, and Alexander Hulpke.