Forked Temperley–Lieb algebras and intermediate subfactors

We consider noncommuting pairs P, Q of intermediate subfactors of an irreducible, finite-index inclusion N⊂M of II1 factors such that P and Q are supertransitive with Jones index less than 4 over N. We show that up to isomorphism of the standard invariant, there is a unique such pair corresponding to each even value but none for the odd values . We also classify the angle values which occur between pairs of intermediate subfactors with small index over their intersection: if [P:N],[Q:N]<4, then the unique nontrivial angle value is always .