Decorated Merge Trees for Persistent Topology

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This paper introduces decorated merge trees (DMTs) as a novel invariant for persistent spaces.

• DMTs combine both $\pi_0$ and $H_n$ information into a single data structure that distinguishes filtrations that merge trees and persistent homology cannot distinguish alone.
• Three variants on DMTs, which emphasize category theory, representation theory and persistence barcodes, respectively, offer different advantages in terms of theory and computation.
• Two notions of distance -- an interleaving distance and bottleneck distance -- for DMTs are defined and a hierarchy of stability results that both refine and generalize existing stability results is proved here.