  Not all planar algebras can encode the algebraic structure of a Yang--Baxter
integrable model described in terms of a so-called homogeneous transfer
operator. In the family of subfactor planar algebras, we focus on the ones
known as singly generated and find that the only such planar algebras
underlying homogeneous Yang--Baxter integrable models are the so-called
Yang--Baxter relation planar algebras. According to a result of Liu, there are
three such planar algebras: the well-known Fuss--Catalan and
Birman--Wenzl--Murakami planar algebras, in addition to one more which we refer
to as the Liu planar algebra. The Fuss--Catalan and Birman--Wenzl--Murakami
algebras are known to underlie Yang--Baxter integrable models, and we show that
the Liu algebra likewise admits a Baxterisation. We also show that the
homogeneous transfer operator describing a model underlied by a singly
generated Yang--Baxter relation planar algebra is polynomialisable, meaning
that it is polynomial in a spectral-parameter-independent element of the
algebra.
