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A complete characterization of the one-dimensional intrinsic Čech persistence diagrams for metric graphs. (English) Zbl 1422.55037
Chambers, Erin Wolf (ed.) et al., Research in computational topology. Based on the first workshop for women in computational topology, Minneapolis, MN, USA, August 2016. Cham: Springer; Minneapolis, MN: Institute for Mathematics and its Applications (IMA). Assoc. Women Math. Ser. 13, 33-56 (2018).
This paper characterizes the 1-dimensional persistent homology of intrinsic Čech complexes of a metric graph \(G\) with a finite number of loops. Indeed, if the metric graph \(G\) is homotopy equivalent to a wedge sum of \(n\) circles, then the 1-dimensional persistent homology barcode has \(n\) intervals, each with birth time zero. The death times of the intervals are given by the lengths of a lexicographically shortest basis for 1-dimensional homology of \(G\) (divided by four). The algebraic arguments behind the proof are carefully constructed, and the figures illustrate the main ideas.
The result is well-motivated by applications of metric graphs to biology, to social networks, and to transportation systems. The paper has motivated extensions from metric graphs to geodesic spaces [Ž. Virk, Rev. Mat. Complut. 32, No. 1, 195–213 (2019; Zbl 1412.55018)]. The paper also advertises an interesting open question: the higher-dimensional persistent homology of a single loop is given in [M. Adamaszek and H. Adams, Pac. J. Math. 290, No. 1, 1–40 (2017; Zbl 1366.05124)], and the 1-dimensional persistent homology of general metric graphs is given in this paper, but what can be said about the higher-dimensional persistent homology of general metric graphs?
For the entire collection see [Zbl 1401.55001].

MSC:
55U10 Simplicial sets and complexes in algebraic topology
05E45 Combinatorial aspects of simplicial complexes
68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
55N35 Other homology theories in algebraic topology
51F99 Metric geometry
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