Composed inclusions of A3 and A4 subfactors

In this article, we classify all standard invariants that can arise from a composed inclusion of an A3 with an A4 subfactor. More precisely, if N⊂P is an A3 subfactor and P⊂M is an A4 subfactor, then only four standard invariants can arise from the composed inclusion N⊂M. We answer a question posed by Bisch and Haagerup in 1994. The techniques of this paper also show that there are exactly four standard invariants for the composed inclusion of two A4 subfactors.