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Merge Trees of Periodic Filtrations
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abstract
Motivated by applications to crystalline materials, we generalize the merge tree and the related barcode of a filtered complex to the periodic setting in Euclidean space. They are invariant under isometries, changing bases, and indeed changing lattices. In addition, we prove stability under perturbations and provide an algorithm that under mild geometric conditions typically satisfied by crystalline materials takes $\mathcal{O}({(n+m) \log n})$ time, in which $n$ and $m$ are the numbers of vertices and edges in the quotient complex, respectively.
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Cited by 2 Pith papers
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Persistent Local Systems of Periodic Spaces
Toroidal cycles of periodic cell complexes embed into canonical persistent local systems, enabling a proposed classification and polynomial-time computation for arbitrary periodicity.
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Identifying cobordisms using kernel persistence
A new homological definition of open cobordisms, or tunnels, connecting two subcomplexes, with a matrix reduction algorithm that pairs their birth and death times in a filtration.
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