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

Agustina Czenky; Melody Molander

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arXiv · 2610.40498

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 $\mathrm{Vec}_{\mathrm{D}_{2n}}^ω$. Applying a jellyfish algorithm on our diagrammatics, we obtain an explicit closed formula for a complete set of normalized 3-cocycle representatives of $\mathrm{D}_{2n}$. 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 $\mathrm{Vec}_{\mathrm{D}_{2n}}^ω$.

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BibTeX RIS

Agustina Czenky, Melody Molander. 2026-10-29. Dihedral pointed fusion categories: diagrammatics, 3-cocycle representatives, and planar algebras. https://arxiv.org/abs/2610.40498

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