Skip to content
DraconiticResearch
Search Sources API
181,686 studies31 sourcesUpdated 52 min ago
PreprintCC BY-NC-ND 4.0arXiv

Higher Barbell Diffeomorphisms and Watanabe's Clasper Surgery

  • Xiayu Tan
Open at sourcePDFDOICite

Abstract

Watanabe constructed many nontrivial family diffeomorphisms in π*Diff∂(D4), 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 πk−1Diff∂(Mk+1') with Mk+1'=naturalk+1 S2× D2 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 D4 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.

Related

  1. Barbell twists are natural
    Yi Liu
    arXivPreprintCC BY 4.0Full textGeometry & topologyarXiv
  2. Higher smooth surgery structure sets of complex projective spaces, part II
    Samuel Kalužný, Tibor Macko
    arXivPreprintCC BY 4.0Full textGeometry & topologyarXiv
  3. Watanabe's Learning Coefficients in Item Response Theory
    Joe Suzuki, Takumi Ikejiri, Yuki Kurumadani
    arXivPreprintarXiv licenceFull textStatisticsarXiv
  4. Systems of rough stochastic differential equations I: weak existence and Yamada--Watanabe
    Florian Huber
    arXivPreprintarXiv licenceFull textProbabilityarXiv
  5. Codimension-one holomorphic Anosov diffeomorphisms
    Jiesong Zhang
    arXivPreprintarXiv licenceFull textAnalysis & PDEsDynamical systemsarXiv

Cite

BibTeXRIS
BibTeX
@misc{tan2026higher,
  title = {{Higher Barbell Diffeomorphisms and Watanabe's Clasper Surgery}},
  author = {Xiayu Tan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24697},
  eprint = {2609.24697},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24697},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Higher Barbell Diffeomorphisms and Watanabe's Clasper Surgery
AU  - Xiayu Tan
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24697
UR  - https://arxiv.org/abs/2609.24697
AB  - Watanabe constructed many nontrivial family diffeomorphisms in π_*Diff_∂(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−1Diff_∂(M_k+1') with M_k+1'=natural_k+1 S^2× 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.
ER  -