{
  "abstract": "Furusho, Hirose and Komiyama reformulated the pentagon equation using an involution called the flip, which led to a new characterisation of the graded Grothendieck-Teichmüller group $\\mathsf{GRT}_1$. We show that the flip is a reflection in the cyclic structure of the operad of framed Drinfeld-Kohno Lie algebras, with the framings removed. Applying the reflection twice to the pentagon gives the two-cycle relation for every Drinfeld associator. This relation is precisely the condition for cyclic compatibility on the associativity morphism, giving a new proof that $\\mathsf{GRT}_1$ acts by cyclic automorphisms of parenthesized ribbon chord diagrams.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17482",
  "author_total": 1,
  "authors": [
    {
      "name": "Chandan Singh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9dpb9p7h05m32.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17482",
  "fulltext_url": null,
  "id": "9dpb9p7h05m32",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17482",
  "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"
    ],
    [
      "arxiv",
      "math.QA"
    ],
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      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
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  "openalex": "W7221774112",
  "pdf_url": "https://arxiv.org/pdf/2610.17482",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9dpb9p7h05m32.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17482",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17482"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Cyclic reflections and the flip equation for the graded Grothendieck-Teichmüller group",
  "title_text": "Cyclic reflections and the flip equation for the graded Grothendieck-Teichmüller group",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/9dpb9p7h05m32",
  "venue": "arXiv",
  "versions": []
}