Essential Simplices Dominate in Harmonic Representatives of One-Dimensional Persistent Classes

Saugata Basu, Aldo Guzmán-Sáenz, Laxmi Parida

arXiv:2607.26378v1 Primary: math.AT Secondary: math.CO

Abstract

Persistent homology summarizes the birth and death of topological features, but it does not by itself specify where a feature is located in the underlying complex. Harmonic persistent homology addresses this by assigning canonical harmonic cycle representatives to bars. In earlier work, Basu and Cox showed that harmonic representatives of simple bars maximize the total relative weight placed on essential simplices, the simplices that are forced to appear in representatives of the corresponding class. In this paper we prove that, for generic one-dimensional bars, this preference is stronger than an aggregate maximization statement. Every essential edge has strictly larger coefficient, in absolute value, than every non-essential edge in the harmonic representative, and the absolute values of the coefficients of all essential edges are equal. The key argument is a finite-dimensional variational characterization of the harmonic representative as a minimum-norm chain with prescribed boundary, combined with an elementary graph-theoretic cut argument. We then prove that the result is special to dimension one. In higher dimensions, the analogous coefficient-wise dominance statement fails. We give examples to show that harmonic representatives can place larger coefficients on non-essential higher-dimensional simplices than on essential ones. These results clarify both the power and the limitations of using harmonic representatives to assign geometric significance to simplices in persistent homology.

Disclosure

“r in practical applications and empirically check the validity of this observation. One first step in this direction was taken in [10]. Acknowledgements We acknowledge the use of the AI programs Chatgpt and Claude for assistance with exploring counterexamples during the brainstorming phase, as well as for reviewing the manuscript and of- fering suggestions during the paper-writing phase. All mathematical proofs, derivations, and final ver”
PDF page 28

Classification: Brainstorming or outlining
Multiplier: 2
Verified: 2026-08-23T19:11:47.598476Z

Structural counts

Pages 29 pdf

Theorems 5 source

Lemmas 9 source

Propositions 6 source

Corollaries 0 source

Definitions 18 source

Displayed equations 225 source

Bibliography entries 84 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file main.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.