# Topological Persistence for Circle-Valued Maps

@article{Burghelea2013TopologicalPF, title={Topological Persistence for Circle-Valued Maps}, author={Dan Burghelea and Tamal K. Dey}, journal={Discrete \& Computational Geometry}, year={2013}, volume={50}, pages={69-98} }

We study circle-valued maps and consider the persistence of the homology of their fibers. The outcome is a finite collection of computable invariants which answer the basic questions on persistence and in addition encode the topology of the source space and its relevant subspaces. Unlike persistence of real-valued maps, circle-valued maps enjoy a different class of invariants called Jordan cells in addition to bar codes. We establish a relation between the homology of the source space and of…

## 36 Citations

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This paper proposes a practical algorithm for computing persistence under Z2 coefficients for a (monotone) sequence of general simplicial maps and shows how these maps arise naturally in some applications of topological data analysis.

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New computable invariants, the "relevant level persistence numbers” and the “positive and negative bar codes” are introduced and explained, and how they are related to the bar codes for level persistence are explained.

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It is shown that the persistence for height functions on general simplicial complexes K linearly embedded in R, hence called height persistence, can be computed in O(n log n) time, which improves significantly the current best bound of O( n), ω being the exponent of matrix multiplication.

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- Computer ScienceSODA
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It is shown that the persistence for height functions on general simplicial complexes K linearly embedded in R, hence called height persistence, can be computed in O(n log n) time, which improves significantly the current best bound of O( n), ω being the exponent of matrix multiplication.

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