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PreprintCC BY-NC-SA 4.0arXiv

Counting tangles in coverings of closed 2-orbifolds via ribbon categories

  • Adam Klukowski
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Abstract

We describe what a typical covering of a closed 2-orbifold looks like on a small scale. Specifically, we prove that a uniformly random covering of degree n contains in expectation Θ(n0) short closed geodesics, potentially Θ(n0) pairs of nearby order-2 cone points, in general Θ(n1/m) cone points of order m, and with high probability no small regions with more complicated topology. This generalises many results of Magee and Puder (2023) from orientable surfaces to possibly non-orientable orbifolds, and answers some of the questions raised by Puder and Zimhoni (2024).

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Cite

BibTeXRIS
BibTeX
@misc{klukowski2026counting,
  title = {{Counting tangles in coverings of closed 2-orbifolds via ribbon categories}},
  author = {Adam Klukowski},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23652},
  eprint = {2609.23652},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23652},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Counting tangles in coverings of closed 2-orbifolds via ribbon categories
AU  - Adam Klukowski
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23652
UR  - https://arxiv.org/abs/2609.23652
AB  - We describe what a typical covering of a closed 2-orbifold looks like on a small scale. Specifically, we prove that a uniformly random covering of degree n contains in expectation Θ(n^0) short closed geodesics, potentially Θ(n^0) pairs of nearby order-2 cone points, in general Θ(n^1/m) cone points of order m, and with high probability no small regions with more complicated topology. This generalises many results of Magee and Puder (2023) from orientable surfaces to possibly non-orientable orbifolds, and answers some of the questions raised by Puder and Zimhoni (2024).
ER  -