On the planar algebra of Ocneanu's asymptotic inclusion

In recent joint work with V. Jones and D. Shlyakhtenko, we have given a diagrammatic description of Popa's symmetric enveloping inclusion for planar algebra subfactors. In this paper we give a diagrammatic construction of the associated Jones tower, in the case that the planar algebra is finite-depth. We then use this construction to describe the planar algebra of the symmetric enveloping inclusion, which is known to be isomorphic to the planar algebra of Ocneanu's asymptotic inclusion by a result of Popa. As an application we give a planar algebraic computation of the (reduced) fusion algebra of the asymptotic inclusion, recovering some well-known results of Ocneanu and Evans-Kawahigashi.