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

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 $\mathrm{Vec}_{\mathrm{D}_{2n}}^ω$.