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PreprintCC BY-NC-SA 4.0arXiv

Dihedral pointed fusion categories: diagrammatics, 3-cocycle representatives, and planar algebras

  • Agustina Czenky
  • Melody Molander
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Abstract

We construct a diagrammatic presentation by generators and relations for all dihedral pointed fusion categories VecD2nω. Applying a jellyfish algorithm on our diagrammatics, we obtain an explicit closed formula for a complete set of normalized 3-cocycle representatives of D2n. As applications of our diagrammatics, we prove a universal property and show that in some cases we can recover affine A unshaded subfactor planar algebras. We then extend our presentation to obtain diagrammatics for all pivotal structures on VecD2nω.

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Cite

BibTeXRIS
BibTeX
@misc{czenky2026dihedral,
  title = {{Dihedral pointed fusion categories: diagrammatics, 3-cocycle representatives, and planar algebras}},
  author = {Agustina Czenky and Melody Molander},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40498},
  eprint = {2610.40498},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40498},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Dihedral pointed fusion categories: diagrammatics, 3-cocycle representatives, and planar algebras
AU  - Agustina Czenky
AU  - Melody Molander
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40498
UR  - https://arxiv.org/abs/2610.40498
AB  - We construct a diagrammatic presentation by generators and relations for all dihedral pointed fusion categories Vec_D_2n^ω. Applying a jellyfish algorithm on our diagrammatics, we obtain an explicit closed formula for a complete set of normalized 3-cocycle representatives of D_2n. As applications of our diagrammatics, we prove a universal property and show that in some cases we can recover affine A unshaded subfactor planar algebras. We then extend our presentation to obtain diagrammatics for all pivotal structures on Vec_D_2n^ω.
ER  -