Title: Barcode Embeddings for Metric Graphs,
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 define a new topology on the space of metric graphs, which we call the
fibered topology, for which the barcode transform is injective on a generic
(open and dense) subset.