Computation with Degree-Rips Bifiltrations
Wednesday, August 15, 2018 - 1:00pm - 1:30pm
RIVET computes degree-Rips bifiltrations, a two-parameter version of the Vietoris Rips complex that takes the density of points into account. In a Rips complex, a simplex appears when all of its faces appear. When computing single parameter persistent homology, we can simply take the maximum time of appearance of its faces to be the time of appearance of a particular simplex. However, in the two-parameter case, the picture is complicated because a simplex can appear at multiple incomparable indices. In addition, a simplex having multiple grades of appearance causes the graded module structure to potentially no longer be free. In this talk, we present an algorithm that computes these grades of appearance in an inductive manner and also present a trick due to Chacholski et al. to get new free modules whose boundary maps give rise to the same homology group.