We study unitary quotients of the free product unitary pivotal category A
2 * T
2. We show 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 T
2 * T
2, 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 A
2 × T
2 and T
2 . T
2, 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 A
3 and A
4 subfactor planar algebras (arxiv:1308.5691). In 1994, BischHaagerup showed that the principal graph of a composite of A
3 and A
4 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.
This is an abridged version of arxiv:1308.5723.