The Fiber of the Persistence Map for Functions on the Interval

In this paper we study functions on the interval that have the same persistent homology. By introducing an equivalence relation modeled after topological conjugacy, which we call graph-equivalence, a precise enumeration of functions with the same persistent homology is given, inviting comparisons with Arnold's Calculus of Snakes. The equivalence classes used here are indexed by chiral merge trees, which are binary merge trees where a left-right ordering of the children of each vertex is given. Enumeration of merge trees and chiral merge trees with the same persistence makes essential use of the Elder Rule (a criterion for pairing critical points), which is given a new proof here as well.