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.

中文翻译：

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Thurston 关系子因子平面代数的分类

Bisch 和 Jones 提出了平面代数的绞线理论分类，并与第二作者一起研究了由 2-boxes 生成的那些。在本文中，我们考虑 3-box 生成器并对由满足 Thurston 提出的关系的非平凡 3-box 生成的子因子平面代数进行分类。分类中的子因子平面代数是 $E^6$ 或来自量子 $SU(N)$ 表示的那些。我们引入了一种确定平面代数正性的新方法和降低计算复杂性的新技术。