Just for fun, I set up the following scheme:  

- A 6j symbol is everything that fulfils Biedenharn-Elliott. (Plus symmetry, orthogonality etc. if that doesn't follow from it anyway.)  

- There are only a finite number $i$ of irreps (including $1$, the 1-dim "neutral").
Then I tried out which of the many possible Clebsch-Gordan expansions are mutally consistent with all the 6j equations via Biedenharn-Elliott. For two irreps I got only this: $1\bigotimes{X}=X (X=1,2), 2\bigotimes{2}=1+2.$  

- Ha!  

- Even I did see this before, it's the smallest Fibonacci fusion category. Now a quick research gave me a load of adjectives that come with fusion categories. Can you tell me which of them apply to my scheme?  

- $i=3$ produces 3 "minis", with $i=4$ I get 6 (and I stopped here because of the dreaded combinatiorial explosion - are these already classified somewhere?).  

- My main question, though, is whether you get a free parameter when you choose $i$ large enough. All my "minis" give only a set of fixed complex numbers for the values of the 6j symbols, the quantum dimension and the writhe normalizer (the minis all seem to be knot-theory compatible). The latter is a root of unity so I can speculate all the minis correspond to special values of the Jones polynomial. Or suchlike.