The Depth Poset under Transpositions in the Filter

TL;DR

This paper investigates how transpositions in filtered Lefschetz complexes affect the depth poset, providing algorithms, case analysis, and statistical insights into the dependencies between birth-death pairs in persistence computations.

Contribution

It introduces fast algorithms for computing and updating the depth poset under transpositions, along with a comprehensive case analysis and statistical analysis on random data.

Findings

Transpositions significantly influence the depth poset structure.

Algorithms enable efficient updates of persistence diagrams under transpositions.

Statistical analysis reveals sensitivity of the depth poset to random transpositions.

Abstract

The depth poset of a filtered Lefschetz complex reflects the dependencies between the cancellations of different shallow birth-death pairs. Using the fast algorithms for computing the depth poset in the present work and for updating the persistence diagram under transpositions (Vineyard persistence), we give a complete case analysis of how transpositions of cells in the filter affect the depth poset. In addition, we present statistics on the depth poset for random point data and its sensitivity to the transpositions that occur in random straight-line homotopies.