Barcode Embeddings for Metric Graphs

Understand

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.