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, 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.

[1]  Michael Lesnick,et al.  The Theory of the Interleaving Distance on Multidimensional Persistence Modules , 2011, Found. Comput. Math..

[2]  R. Ghrist.  Barcodes: The persistent topology of data , 2007 .

[3]  Herbert Edelsbrunner,et al.  Computational Topology - an Introduction , 2009 .

[4]  W. Crawley-Boevey.  Decomposition of pointwise finite-dimensional persistence modules , 2012, 1210.0819.

[5]  J. Curry.  Topological data analysis and cosheaves , 2014, 1411.0613.

[6]  Ronald L. Rivest,et al.  Introduction to Algorithms, third edition , 2009 .

[7]  Vin de Silva,et al.  Metrics for Generalized Persistence Modules , 2013, Found. Comput. Math..

[8]  Afra Zomorodian,et al.  Computing Persistent Homology , 2004, SCG '04.

[9]  Steve Oudot,et al.  The Structure and Stability of Persistence Modules , 2012, Springer Briefs in Mathematics.

[10]  Bayoumi I. Bayoumi,et al.  Counting two-dimensional posets , 1994, Discret. Math..

[11]  R. Ghrist,et al.  Euler Calculus with Applications to Signals and Sensing , 2012, 1202.0275.

[12]  Justin Curry,et al.  Classification of Constructible Cosheaves , 2016, 1603.01587.

[13]  Vin de Silva,et al.  Theory of interleavings on categories with a flow , 2017, 1706.04095.

[14]  David Cohen-Steiner,et al.  Stability of Persistence Diagrams , 2005, Discret. Comput. Geom..

[15]  Amit Patel,et al.  Generalized persistence diagrams , 2016, J. Appl. Comput. Topol..

[16]  P. Gabriel.  Unzerlegbare Darstellungen I , 1972 .

[17]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[18]  Pierre Schapira,et al.  Operations on constructible functions , 1991 .

[19]  Herbert Edelsbrunner,et al.  Topological persistence and simplification , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[20]  Gunnar E. Carlsson,et al.  Topology and data , 2009 .

[21]  Vladimir I. Arnold,et al.  The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups , 1992 .