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Quantum Topology


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Volume 10, Issue 3, 2019, pp. 441–480
DOI: 10.4171/QT/126

Published online: 2019-09-03

Classification of Thurston relation subfactor planar algebras

Corey Jones[1], Zhengwei Liu[2] and Yunxiang Ren[3]

(1) Ohio State University, Columbus, USA
(2) Harvard University, Cambridge, USA
(3) Harvard University, Cambridge, USA

Bisch and Jones proposed the classification of planar algebras by simple generators and relations. They investigated with the second author the classification of planar algebras generated by 2-boxes. In this paper, we classify singly-generated Thurston-relation planar algebras, defined as subfactor planar algebras generated by a 3-box satisfying a relation proposed by Dylan Thurston. Our main result shows that such subfactor planar algebras are either the $E_6$ subfactor planar algebras or belong to a two-parameter family of planar algebras arising from the representations of type $A$ quantum groups. We introduce a new method for determining positivity of the Markov trace of planar algebras in this family.

Keywords: Subfactors, planar algebras, Thurston relation, skein theory

Jones Corey, Liu Zhengwei, Ren Yunxiang: Classification of Thurston relation subfactor planar algebras. Quantum Topol. 10 (2019), 441-480. doi: 10.4171/QT/126