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On unifying Jones representations of braid groups and Thompson's group

  • Yangxiao Luo
  • Shunyu Wan
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Abstract

We unify Jones' representations of braid groups and the oriented Thompson's group by introducing the oriented dyadic braid group. This finitely generated group contains every braid group and the oriented Thompson's group. We construct an embedding of the oriented Thompson's group into this group such that taking braid closures recovers Jones' construction of oriented links. We also introduce the dyadic Temperley-Lieb algebra and construct a unitary representation of the oriented dyadic braid group extending the classical Jones representations of braid groups. Along the Thompson group's embedding, its distinguished matrix coefficient agrees with the corresponding coefficient of Jones' representation of the oriented Thompson's group.

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Cite

BibTeXRIS
BibTeX
@misc{luo2026unifying,
  title = {{On unifying Jones representations of braid groups and Thompson's group}},
  author = {Yangxiao Luo and Shunyu Wan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.09333},
  eprint = {2610.09333},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.09333},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - On unifying Jones representations of braid groups and Thompson's group
AU  - Yangxiao Luo
AU  - Shunyu Wan
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.09333
UR  - https://arxiv.org/abs/2610.09333
AB  - We unify Jones' representations of braid groups and the oriented Thompson's group by introducing the oriented dyadic braid group. This finitely generated group contains every braid group and the oriented Thompson's group. We construct an embedding of the oriented Thompson's group into this group such that taking braid closures recovers Jones' construction of oriented links. We also introduce the dyadic Temperley-Lieb algebra and construct a unitary representation of the oriented dyadic braid group extending the classical Jones representations of braid groups. Along the Thompson group's embedding, its distinguished matrix coefficient agrees with the corresponding coefficient of Jones' representation of the oriented Thompson's group.
ER  -