A fusion category is called transparent if the associator involving any invertible object is the identity map. For the Haagerup-Izumi fusion rings with G=Z2n+1 (the Z3 case is the Haagerup fusion ring with six simple objects), the transparent ansatz reduces the number of independent F
Nearby in the stack
-symbols from order
O(n6)
to
O(n2)
, rendering the pentagon identity practically solvable. Transparent Haagerup-Izumi fusion categories are thereby constructively classified up to
G=Z9
, recovering all known Haagerup-Izumi fusion categories to this order, and producing new ones. Transparent Haagerup-Izumi fusion categories additionally satisfying
S4
tetrahedral invariance are further classified up to
G=Z15
, and the explicit
F
-symbols for the unitary ones, including the Haagerup
H3
fusion category, are compactly presented. The
F
-symbols for the Haagerup
H2
fusion category are also presented. Going beyond, the transparent ansatz offers a viable course towards constructing novel fusion categories for new fusion rings.