We study unitary quotients of the free product unitary pivotal category
. We show that such quotients are parametrized by an integer
–th root of unity
. We show that for
, there is exactly one quotient and
$4\,\le \,n\,\le \,10$
, we show that there are no such quotients. Our methods also apply to quotients of
, 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
, 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
subfactor planar algebras (arxiv:1308.5691). In 1994, Bisch–Haagerup showed that the principal graph of a composite of
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
This is an abridged version of arxiv:1308.5723.