  Jones and Penneys showed that a finite depth subfactor planar algebra embeds
in the bipartite graph planar algebra of its principal graph, via a Markov
towers of algebras approach. We relate several equivalent perspectives on the
notion of module over a subfactor planar algebra, and show that a Markov tower
is equivalent to a module over the Temperley-Lieb-Jones planar algebra. As a
corollary, we obtain a classification of semisimple pivotal C* modules over
Temperley-Lieb-Jones in terms of pointed graphs with a Frobenius-Perron vertex
weighting. We then generalize the Markov towers of algebras approach to show
that a finite depth subfactor planar algebra embeds in the bipartite graph
planar algebra of the fusion graph of any of its cyclic modules.
