A quadrilateral in the Asaeda–Haagerup category

Abstract

We construct a noncommuting quadrilateral of factors whose upper sides are each the Asaeda–Haagerup subfactor with index $\frac{5+\sqrt{17}}{2}$ by showing the existence of a $Q$-system in the Asaeda–Haagerup category with index $\frac{7+\sqrt{17}}{2}$ . We also conjecture the existence of a Q-system in the same category with index $\frac{9+\sqrt{17}}{2}$ and an associated quadrilateral whose upper sides have index $\frac{7+\sqrt{17}}{2}$ .