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PreprintCC BY 4.0arXiv

Quasipositive Braids, Presentations, and Garside Structures

  • Jacob Guynee
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Abstract

We provide a presentation of the braid group in which the generating set is the set of all positive half-twists, the relations are balanced positive relations, and every relation is conjugate to one in a finite list. We then prove that for any such presentation, its positive monoid is not isomorphic to the quasipositive braid monoid. Finally, we show that the quasipositive braid monoid has no proper Garside family. These results settle a problem of Calegari from the K3 problems list.

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Cite

BibTeXRIS
BibTeX
@misc{guynee2026quasipositive,
  title = {{Quasipositive Braids, Presentations, and Garside Structures}},
  author = {Jacob Guynee},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12808},
  eprint = {2610.12808},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12808},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Quasipositive Braids, Presentations, and Garside Structures
AU  - Jacob Guynee
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12808
UR  - https://arxiv.org/abs/2610.12808
AB  - We provide a presentation of the braid group in which the generating set is the set of all positive half-twists, the relations are balanced positive relations, and every relation is conjugate to one in a finite list. We then prove that for any such presentation, its positive monoid is not isomorphic to the quasipositive braid monoid. Finally, we show that the quasipositive braid monoid has no proper Garside family. These results settle a problem of Calegari from the K3 problems list.
ER  -