A Directed Cubical Framework for k-critical Bipersistence Modules
Abstract
In topological data analysis, multiparameter persistence modules (MPMs) are used to analyze data with multiple parameters. Summarizing topological information for MPMs is a non-trivial task; one useful summary of bipersistence modules is the fibered barcode \cite{lesnick2015interactive,lesnick2025fast}. In this paper, we use a directed topological construction of the parameter space of a bifiltration to extend the fibered barcode. Specifically, given a k-critical bifiltration F, we build an associated directed Euclidean cubical complex C whose directed paths each correspond to a one-parameter subfiltration of F. The path space of C, diC, is used to define the \emph{extended fibered barcode}, which contains all possible monotonically increasing paths in C. We show that there can be paths in diC which generate filtrations not obtainable along a line with positive slope, and that all lines with positive slope have a path in diC that generate an equivalent barcode. We introduce two types of equivalence for these restricted modules: combinatorial equivalence of diagrams, and simplicial equivalence of the filtrations generating the modules. We show that removing certain cubes from C yields a subcomplex on which dihomotopic paths induce simplicially equivalent filtrations, and hence combinatorially equivalent diagrams.
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BibTeX
@misc{belton2026directed,
title = {{A Directed Cubical Framework for k-critical Bipersistence Modules}},
author = {Robin Belton and Robyn Brooks and Lisbeth Fajstrup and Brittany Terese Fasy and Elizabeth Vidaurre},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00261},
eprint = {2611.00261},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00261},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Directed Cubical Framework for k-critical Bipersistence Modules
AU - Robin Belton
AU - Robyn Brooks
AU - Lisbeth Fajstrup
AU - Brittany Terese Fasy
AU - Elizabeth Vidaurre
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00261
UR - https://arxiv.org/abs/2611.00261
AB - In topological data analysis, multiparameter persistence modules (MPMs) are used to analyze data with multiple parameters. Summarizing topological information for MPMs is a non-trivial task; one useful summary of bipersistence modules is the fibered barcode \cite{lesnick2015interactive,lesnick2025fast}. In this paper, we use a directed topological construction of the parameter space of a bifiltration to extend the fibered barcode. Specifically, given a k-critical bifiltration F, we build an associated directed Euclidean cubical complex C whose directed paths each correspond to a one-parameter subfiltration of F. The path space of C, diC, is used to define the \emph{extended fibered barcode}, which contains all possible monotonically increasing paths in C. We show that there can be paths in diC which generate filtrations not obtainable along a line with positive slope, and that all lines with positive slope have a path in diC that generate an equivalent barcode. We introduce two types of equivalence for these restricted modules: combinatorial equivalence of diagrams, and simplicial equivalence of the filtrations generating the modules. We show that removing certain cubes from C yields a subcomplex on which dihomotopic paths induce simplicially equivalent filtrations, and hence combinatorially equivalent diagrams.
ER -