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The Lorenz braid index and hyperbolic volume

  • Thiago de Paiva
  • Connie On Yu Hui
  • José Andrés Rodríguez Migueles
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Abstract

A result of Futer, Kalfagianni, and Purcell implies that an upper volume bound for all link complements in the 3-sphere cannot depend solely on the braid index. In this paper, we introduce the Lorenz braid index and generalise the bunch algorithm to provide a general upper volume bound for all link complements in the 3-sphere. Such an upper bound is a quadratic polynomial in the Lorenz braid index. In addition, we construct an explicit family of hyperbolic Lorenz knots for which the classical braid index and the Seifert genus both tend to infinity, while the Lorenz braid index remains bounded.

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Cite

BibTeXRIS
BibTeX
@misc{paiva2026lorenz,
  title = {{The Lorenz braid index and hyperbolic volume}},
  author = {Thiago de Paiva and Connie On Yu Hui and José Andrés Rodríguez Migueles},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14931},
  eprint = {2609.14931},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14931},
}
RIS
TY  - UNPB
TI  - The Lorenz braid index and hyperbolic volume
AU  - Thiago de Paiva
AU  - Connie On Yu Hui
AU  - José Andrés Rodríguez Migueles
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.14931
UR  - https://arxiv.org/abs/2609.14931
AB  - A result of Futer, Kalfagianni, and Purcell implies that an upper volume bound for all link complements in the 3-sphere cannot depend solely on the braid index. In this paper, we introduce the Lorenz braid index and generalise the bunch algorithm to provide a general upper volume bound for all link complements in the 3-sphere. Such an upper bound is a quadratic polynomial in the Lorenz braid index. In addition, we construct an explicit family of hyperbolic Lorenz knots for which the classical braid index and the Seifert genus both tend to infinity, while the Lorenz braid index remains bounded.
ER  -