The fiber of the persistence map for functions on the interval

Abstract

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. arXiv:1706.06059v2 [math.AT]

Contractibility of a persistence map preimage

“…Recall from Remark 2.6 that Curry ( 2018 ) provides a count of the contractible components of the preimage of a persistence map. However, to the best of our knowledge, this paper provides the first detailed analysis of the homotopy type of these of a components.…”

“…The components C(z) group vectors into equivalence classes that can be characterized using the notion of chiral merge tree as defined in Curry (2018). Corollary 5.5 of Curry (2018) shows that the number of chiral merge trees realizing diagram P is equal to 2…”

“…The components C ( z ) group vectors into equivalence classes that can be characterized using the notion of chiral merge tree as defined in Curry ( 2018 ). Corollary 5.5 of Curry ( 2018 ) shows that the number of chiral merge trees realizing diagram P is equal to , where B is the barcode realization of P , i.e. set of intervals having the birth and death values of the j -th persistence point as its endpoints, and is the number of intervals in B that contain .…”

Scalar Field Comparison with Topological Descriptors: Properties and Applications for Scientific Visualization

“…4(b–c), where a merge tree decomposes into a barcode following a branch decomposition process; and bars in a barcode can be used to assemble a (non‐unique) merge tree following a gluing process. See [CCF∗20, Cur18, KGH20] for references for the relation between a merge tree and a barcode. Note that the notions of join and split trees [CSA03] are the two forms of merge trees; a join tree is the merge tree of f and a split tree is the merge tree of – f .…”

From Trees to Barcodes and Back Again: Theoretical and Statistical Perspectives

“…Start by creating a set A of active vertices, originally set equal to L, and an empty barcode. For each leaf l, the algorithm proceeds recursively along its unique path to the root r. At each branch point b, one applies the standard Elder Rule from topological data analysis [22], removing from A all of the children of b, and adding b to A. One bar is added to the barcode for each child of b except (any one of) the longest.…”