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PreprintCC BY-SA 4.0HAL

Nuancing the uniqueness of q-rationals

  • Perrine Jouteur
  • Olga Paris-Romaskevich
  • Alexander Thomas
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Abstract

We prove uniqueness of q-rational numbers up to conjugacy, using character varieties. Despite the unicity, we exhibit a two-parameter family of deformations of rationals with a modular symmetry. We prove that there are exactly two deformations which deliver the usual q-integers: the original q-rationals defined by Morier-Genoud and Ovsienko, and another new one. Although the new family can be obtained by conjugacy from the old one, new positivity properties appear. In addition, this new family provides a direct computation of the Jones polynomial of rational knots.

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Cite

BibTeXRIS
BibTeX
@misc{jouteur2026nuancing,
  title = {{Nuancing the uniqueness of q-rationals}},
  author = {Perrine Jouteur and Olga Paris-Romaskevich and Alexander Thomas},
  howpublished = {HAL},
  year = {2026},
  month = jul,
  doi = {10.48550/arxiv.2607.22308},
  eprint = {2607.22308},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05717044v2},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - Nuancing the uniqueness of q-rationals
AU  - Perrine Jouteur
AU  - Olga Paris-Romaskevich
AU  - Alexander Thomas
PY  - 2026
DA  - 2026/07/24
PB  - HAL
DO  - 10.48550/arxiv.2607.22308
UR  - https://hal.science/hal-05717044v2
AB  - We prove uniqueness of q-rational numbers up to conjugacy, using character varieties. Despite the unicity, we exhibit a two-parameter family of deformations of rationals with a modular symmetry. We prove that there are exactly two deformations which deliver the usual q-integers: the original q-rationals defined by Morier-Genoud and Ovsienko, and another new one. Although the new family can be obtained by conjugacy from the old one, new positivity properties appear. In addition, this new family provides a direct computation of the Jones polynomial of rational knots.
ER  -