The structure of sectors associated with Longo-Rehren inclusions II. Examples

As an application of the general theory established in the first part, we determine the structure of Longo-Rehren inclusions for several systems of sectors arising from endomorphisms of the Cuntz algebras. The E-6 subfactor and the Haagerup subfactor are included among these examples, and the dual principal graphs and the S and T-matrices for their Longo-Rehren inclusions are obtained. We also construct several new subfactors using endomorphisms of the Cuntz algebras, and determine their tube algebra structure.