Singly generated planar algebras of small dimension, Part III

The first two authors classified subfactor planar algebra generated by a non-trivial 2-box subject to the condition that the dimension of 3-boxes is at most 12 in Part I; 13 in Part II of this series. They are the group planar algebra for $\mathbb{Z}_3$, the Fuss-Catalan planar algebra ; and the group/subgroup planar algebra for $\mathbb{Z}_2\subset \mathbb{Z}_5\rtimes \mathbb{Z}_2$. In the present paper, we extend the classification to 14 dimensional 3-boxes. They are all BMW. Precisely it contains a depth 3 one from quantum $SO(3)$, and a one-parameter family from quantum $Sp(4)$.