In this paper we study functions on the interval that have the same persistent homology, which is what we mean by the fiber of the persistence map. By imposing an equivalence relation called graph-equivalence, the fiber of the persistence map becomes finite and a precise enumeration is given. Graph-equivalence classes 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, which is given its first detailed proof in this paper.
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The first real step forward on this problem happened in conversations with Hans Riess, who was an undergraduate at Duke at the time. It was there that the author developed a method for constructing at least one function realizing any suitable barcode. A full characterization of the fiber of the persistence map was obtained after conversations with Yuliy Baryshnikov at ICERM and with John Harer at Duke. Yuliy Baryshnikov first introduced the concept of a chiral barcode associated to a time series in June 2015 in a talk at a conference on Geometry and Data Analysis at the Stevanovich Center for Financial Mathematics, the author then used the underlying concept of a chiral merge tree to create multiple realizations of a single barcode. Finally, John Harer helped cinch the counting result by pointing out the restriction that the Elder Rule imposed on chiral merge trees. The author believes that the conversations with Yuliy and John provided critical insights at the heart of this paper. Other discussions have benefited subsequent revisions of this paper. Rachel Levanger pointed out a deficiency in an earlier version of Definition 4.1. Elizabeth Munch and Anastasios Stefanou first made the connection between interleavings and certain labeled merge trees that eventually appeared in their work (Munch and Stefanou 2018). The author received several helpful comments from each of the anonymous referees, which helped improve the paper. One comment in particular caused the author to think more carefully about maps of ordered merge trees.
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Curry, J. The fiber of the persistence map for functions on the interval. J Appl. and Comput. Topology 2, 301–321 (2018). https://doi.org/10.1007/s41468-019-00024-z
- Persistent homology
- Inverse problems