Merge Trees of Periodic Filtrations

TL;DR

This paper extends merge trees and barcodes to periodic settings in Euclidean space, ensuring invariance under symmetries, stability under perturbations, and providing an efficient algorithm for crystalline materials.

Contribution

It introduces a generalized merge tree and barcode framework for periodic complexes, with invariance properties and a practical algorithm for crystalline applications.

Findings

Invariance under isometries, basis changes, and lattice modifications.

Stability of the periodic merge trees under perturbations.

An efficient algorithm with near-linear complexity for crystalline materials.

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.