An isometry theorem for persistent homology of circle-valued functions

TL;DR

This paper extends persistent homology to circle-valued functions, introducing a new isometry theorem that links interleaving and bottleneck distances through a geometric model, advancing the mathematical understanding of circle-valued persistence modules.

Contribution

It generalizes barcodes for circle-valued functions and proves an isometry theorem connecting key distances in this new setting.

Findings

Established an isometry theorem for circle-valued persistence modules.

Proposed a geometric model using arcs for generalized barcodes.

Extended interleaving and bottleneck distances to circle-valued functions.

Abstract

This paper explores persistence modules for circle-valued functions, presenting a new extension of the interleaving and bottleneck distances in this setting. We propose a natural generalisation of barcodes in terms of arcs on a geometric model associated to the derived category of quiver representations. The main result is an isometry theorem that establishes an equivalence between the interleaving distance and the bottleneck distance for circle-valued persistence modules.