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-LiebJones 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. (C) 2021 The Author(s). Published by Elsevier Inc.
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