Merge Trees of Periodic Filtrations

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.