{
  "abstract": "We show that the evaluations of combinatorial 1-cocycles defined by Gauss diagrams with a triangle on the canonical loops in the space of long knots are finite type invariants. Using Gauss diagrams, we construct two $\\mathbb{Z}$-valued combinatorial 1-cocycles $β_1, β_2$ and a $\\mathbb{Z}/2\\mathbb{Z}$-valued 1-cocycle $β_3$ on the space of long knots and prove their cocyclicity by verifying their invariance under higher Reidemeister moves coming from the codimension-two singularities of plane curves. We show that they represent genuinely new 1-cohomology classes and compute their pairings with the rotation, rolling, half rolling, bracket and half bracket loops. A key new feature is that $β_1$ and $β_2$ can pair nontrivially with bracket and half-bracket loops. We conjecture that $(α_3^1,β_1,β_2)$ over $\\mathbb{Q}$, and their mod 2 reductions together with $β_3$ over $\\mathbb{Z}/2\\mathbb{Z}$, form bases of degree-one cohomology up to order 4 in the sense of Vassiliev. In addition, we show that the reparametrization loop is homotopic to the rolling loop concatenated with the rotation loop in $\\operatorname{Emb}(S^1, S^3)$. Finally, we give the criteria for a 1-cohomology class in the long knot space to descend to 1-cohomology classes in $\\operatorname{Emb}(S^1, S^3)$ and $\\operatorname{Emb}(S^1, S^3)/\\operatorname{Diff}^{+}(S^1)$. Using these criteria, we show that $β_1$ descends to a nontrivial 1-cohomology class in $\\operatorname{Emb}(S^1,S^3)$, while $β_1 \\bmod 2$ and $β_3$ descend to linearly independent nontrivial 1-cohomology classes in $\\operatorname{Emb}(S^1, S^3)/\\operatorname{Diff}^{+}(S^1)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29946",
  "author_total": 1,
  "authors": [
    {
      "name": "Butian Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bhka3t05pj2bb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29946",
  "fulltext_url": null,
  "id": "bhka3t05pj2bb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29946",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7214332253",
  "pdf_url": "https://arxiv.org/pdf/2609.29946",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bhka3t05pj2bb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29946",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29946"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Pairings of combinatorial 1-cocycles with loops in knot spaces",
  "title_text": "Pairings of combinatorial 1-cocycles with loops in knot spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/bhka3t05pj2bb",
  "venue": "arXiv",
  "versions": []
}