  Bisch and Jones suggested the skein theoretic classification of planar
algebras and investigated the ones generated by 2-boxes with the second author.
In this paper, we consider 3-box generators and classify subfactor planar
algebras generated by a non-trivial 3-box satisfying a relation proposed by
Thurston. The subfactor planar algebras in the classification are either $E^6$
or the ones from representations of quantum $SU(N)$. We introduce a new method
to determine positivity of planar algebras and new techniques to reduce the
complexity of computations.
