Quotients of A₂ ^∗ T₂

Westudy unitary quotients of the free product unitary pivotal category A₂ ^∗ T₂. Weshow that such quotients are parametrized by an integer n ≥ 1 and an 2n-th root of unity ω. We show that for n = 1,2,3, there is exactly one quotient and ω = 1. For 4 ≤ n ≤ 10, we show that there are no such quotients. Our methods also apply to quotients of T₂ ^∗ T₂, where we have a similar result. The essence of our method is a consistency check on jellyfish relations. While we only treat the specific cases of A₂ ^∗ T₂ and T₂ ^∗ T₂, we anticipate that our technique can be extended to a general method for proving the nonexistence of planar algebras with a specified principal graph. During the preparation of this manuscript, we learnt of Liu's independent result on composites of A₃ and A₄ subfactor planar algebras (arxiv:1308.5691). In 1994, Bisch-Haagerup showed that the principal graph of a composite of A₃ and A₄ must fit into a certain family, and Liu has classified all such subfactor planar algebras. We explain the connection between the quotient categories and the corresponding composite subfactor planar algebras. As a corollary of Liu's result, there are no such quotient categories for n ≥ 4.