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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} $.
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Asaeda, M. and Haagerup, U., Exotic subfactors of finite depth with Jones indices ( 5 + 13 ) / 2 (5+\sqrt{13})/2 and ( 5 + 17 ) / 2 (5+\sqrt{17})/2 . Comm. Math. Phys
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Grossman, P., and Izumi, M. (2008). Classification of noncommuting quadrilaterals of factors. International Journal of Mathematics
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Jones, V. F. R. (2000). The planar algebras of a bipartite graph. in Knots in Hellas ’98
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Xu, F. (2009). On representing some lattices as lattices of intermediate subfactors of finite index. Advances in Mathematics
2009
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