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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)?

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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.

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