Somewhat nebulous question: there are many well known "special" values of the Jones polynomial, especially those at roots of unity. I always run into one that has unlink value $\phi$ (golden mean) and writhe factor $(1)^{1/5}$. Is there something special about it (maybe it's "at the intersection" of the Lie groups A1 and G2 or whatnot)?
In some sense this is the smallest possible quantum group, so it's perhaps not surprising that it comes up often. In fact, if you have only 2 objects, then there are very few possibilities, see Ostrik's paper http://arxiv.org/abs/math/0203255. 

