{
  "abstract": "Vietoris-Rips filtrations, which are standard in topological data analysis, are flag complexes, and a simplex tree stores these without any attaching data. In this paper we ask what survives of this economy when a flag complex $K$ is divided by a subcomplex $A$, each connected component of $A$ being crushed to a point. Such a quotient is a CW complex whose cells are the simplices of $K\\setminus A$, but their attaching maps are no longer implicit. We show that for flag $K$ the face order of the quotient is strictly graded exactly when $A$ is flag, and that the surviving labelled cells are determined by those of dimension at most 3. For $m$-flag pairs the threshold is $2m+1$, and it drops to $m+2$ when $K$ is flag. The prescribed cells form a regular CW decomposition only when $A$ is full in $K$. These results justify the QF-tree: a cell table that stores, for each surviving simplex, its ordered list of $d+1$ facets with collapsed facets flagged, indexed by a trie of quotient-vertex words. For bounded dimension its size is linear in the number of surviving simplices plus the retained provenance, and we derive and verify a simple formula for the collapsed fraction above which it is smaller than the homotopy-equivalent cone model. Because a collapse changes the attaching data only on the closed star of $A$, the QF-tree can also be applied locally inside a simplex tree. For a ball-shaped $A$ in the sampled Vietoris--Rips regime the closed star is a thin shell, and the median compact budget is below the cone model at every sampled radius. An accompanying library, modelled on Gudhi, implements the QF-tree, its local variant, an editable layer with local quotient updates, gluing, disc attachment, induced maps, cup products, fundamental-group presentations and zigzag persistence, and provided experiments separate the cost of maintaining a quotient from the cost of the algebra computed on it.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32733",
  "author_total": 4,
  "authors": [
    {
      "name": "Konstantin Sorokin"
    },
    {
      "name": "Aleksandr Levin"
    },
    {
      "name": "Maxim Beketov"
    },
    {
      "name": "Anton Ayzenberg"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cybctkzx1w22n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32733",
  "fulltext_url": null,
  "id": "cybctkzx1w22n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32733",
  "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",
      "cs.CG"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.AT"
    ],
    [
      "openalex",
      "1703|Topological and Geometric Data Analysis"
    ]
  ],
  "openalex": "W7214751512",
  "pdf_url": "https://arxiv.org/pdf/2609.32733",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cybctkzx1w22n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32733",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32733"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A data structure for quotient flag complexes",
  "title_text": "A data structure for quotient flag complexes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/cybctkzx1w22n",
  "venue": "arXiv",
  "versions": []
}