{
  "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 $C$, $\\di{C}$, 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 $\\di{C}$ which generate filtrations not obtainable along a line with positive slope, and that all lines with positive slope have a path in $\\di{C}$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00261",
  "author_total": 5,
  "authors": [
    {
      "name": "Robin Belton"
    },
    {
      "name": "Robyn Brooks"
    },
    {
      "name": "Lisbeth Fajstrup"
    },
    {
      "name": "Brittany Terese Fasy"
    },
    {
      "name": "Elizabeth Vidaurre"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7fqmesqf63ztf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00261",
  "fulltext_url": null,
  "id": "7fqmesqf63ztf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00261",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00261",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7fqmesqf63ztf.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00261",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00261"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A Directed Cubical Framework for k-critical Bipersistence Modules",
  "title_text": "A Directed Cubical Framework for k-critical Bipersistence Modules",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/7fqmesqf63ztf",
  "venue": "arXiv",
  "versions": []
}