In standard persistent homology, a persistent cycle born and dying with apersistence interval (bar) associates the bar with a concrete topologicalrepresentative . However, topological features usually go through variations in the lifecycle of a cycle which a single persistent cycle may not capture . We propose levelset persistent cyclesconsisting of a sequence of cycles that depict the evolution of homologicalfeatures from birth to death . Our definition is based on levelset zigzagpersistence which involves four types of persistence intervals as opposed to the two types in standard persistence . For each of the four types, we present apolynomial-time algorithm computing an optimal sequence of levelset persists$p$-cycles for the so-called weak $(p+1)$-pseudomanifolds . Our results are useful in practice because weak pseudomanifils do appear in applications. Our algorithms draw upon an idea of relating optimal cycles to min-cuts in a graph that weexploited earlier for standard persistent cycles in this case

Author(s) : Tamal K. Dey, Tao Hou

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Keywords : persistent - cycles - levelset - cycle - types -

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