Summary: We show that a certain natural class of tangles ‘generate the collection of all tangles with respect to composition’. This result is motivated by, and describes the reasoning behind, the ‘uniqueness assertion’ in Jones’ theorem on the equivalence between extremal subfactors of finite index and what we call ‘subfactor planar algebras’ here. This result is also used to identify the manner in which the planar algebras corresponding to \(M\subset M_1\) and \(N^{op} \subset M^{op}\) are obtained from that of \(N \subset M\).
Our results also show that ‘duality’ in the category of extremal subfactors of finite index extends naturally to the category of ‘general’ planar algebras (not necessarily finite-dimensional or spherical or connected or \(C^{\ast}\), in the terminology of Jones).