Draw a colored loop. You explained quantum dimension. :-) OK, now you may use trivalent nodes and knot crossings too (with edges colored by irreps). You can now pictorialize 6j symbols, writhe normalizers, structure constants etc. My question is whether other things that also pop up all the time in Lie groups, as the (quadratic) Casimir or Dynkin index of an irrep can be visualized this way. (Since that's the only way I can do math :-)

Background: a) In "Birdtracks" an additive formula for Dynkin indices (multiplied with dimensions somehow) of the irreps involved in a Clebsch-Gordan expansion is given, b) I deduced another formula involving writhe normalizers and dimensions that is also valid for any Clebsch-Gordan expansion. (In short: Let $R_i\bigotimes{R_j}=\bigoplus_k{R_k}$, and $W_k=w_i*w_j/w_k$, where the $w_k$ are writhe normalizers, let $o_k$ be the dimensions, then $\Sigma_k{(W_k-W_k^{-1})o_k}=0$. Using pictures the proof is an one-liner.) It's natural to speculate whether a) and b) are essentially the same (or maybe even more things are additive over a Clebsch-Gordan expansion).