Singly Generated Planar Algebras of Small Dimension

0. Introduction. A subfactorN ⊂M gives rise to a powerful set of invariants that can be approached successfully in several ways. (See, for instance, [B2], [EK], [FRS], [GHaJ], [H], [Iz], [JSu], [Lo1], [Lo2], [Oc1], [Oc2], [Po1], [Po2], [Po3], [Po4], [Wa], [We1], and [We2]). A particular approach suggests a particular kind of subfactor as the “simplest.” For instance, in Haagerup’s approach [H], subfactors of small index are the simplest. In [J2], a pictorial language is developed in which the invariants appear as a graded vector space V = (Vn)n≥0 whose elements can be combined in planar, but otherwise quite arbitrary, ways. Thus, for instance, in the diagram