{
  "abstract": "Watanabe constructed many nontrivial family diffeomorphisms in $π_*\\text{Diff}_\\partial(D^4)$, which were obtained by doing clasper surgeries on trivalent graphs. In this paper, we generalize the barbell diffeomorphism discovered by Budney and Gabai and construct a series of nontrivial elements in $π_{k-1}\\text{Diff}_\\partial(M_{k+1}')$ with $M_{k+1}'=\\natural_{k+1} S^2\\times D^2$ the $(k+1)$-cuff barbell, which we call higher barbell diffeomorphisms. We show that Watanabe's constructions can be realized as implanted higher barbell diffeomorphisms in $D^4$ if the trivalent graph is homologically nonzero in the graph complex. Combining this realization with Watanabe's detection theorem, we establish the nontriviality of these implanted higher barbell diffeomorphisms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24697",
  "author_total": 1,
  "authors": [
    {
      "name": "Xiayu Tan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cjef9es4d4bwt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24697",
  "fulltext_url": null,
  "id": "cjef9es4d4bwt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24697",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7214044168",
  "pdf_url": "https://arxiv.org/pdf/2609.24697",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cjef9es4d4bwt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24697",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24697"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Higher Barbell Diffeomorphisms and Watanabe's Clasper Surgery",
  "title_text": "Higher Barbell Diffeomorphisms and Watanabe's Clasper Surgery",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/cjef9es4d4bwt",
  "venue": "arXiv",
  "versions": []
}