Barcode Embeddings for Metric Graphs
Algebraic Topology
2021-08-18 v4 Metric Geometry
Abstract
Stable topological invariants are a cornerstone of persistence theory and applied topology, but their discriminative properties are often poorly-understood. In this paper we study a rich homology-based invariant first defined by Dey, Shi, and Wang, which we think of as embedding a metric graph in the barcode space. We prove that this invariant is locally injective on the space of metric graphs and globally injective on a GH-dense subset. Moreover, we show that is globally injective on a full measure subset of metric graphs, in the appropriate sense.
Cite
@article{arxiv.1712.03630,
title = {Barcode Embeddings for Metric Graphs},
author = {Steve Oudot and Elchanan Solomon},
journal= {arXiv preprint arXiv:1712.03630},
year = {2021}
}
Comments
The newest draft clarifies the proofs in Sections 7 and 8, and provides improved figures therein. It also includes a results section in the introduction