I only ask for the subset of Reshetikhine-Turaev invariants for now.

Is the following sum up into cases a) and b) correct?

- Dots in Dynkin diagrams of Lie groups G come as a) symmetry-equivalent pairs
(most of $G=A_n$, some of $D_n$ and $E_6$), b) not.

- The irreps $R$ belonging to them are a) not self-adjunct b) self-adjunct.

- $R\otimes{R}$ contains a) not 1 b) 1

- The knot polynomial belonging to $R$ is defined for oriented knots anyway, and a) it doesn't work with unoriented ones b) you can as well drop the arrows.

At least this would explain why I can't find a R matrix for $A_2$(dim 3 irrep). (Or more precise, an R matrix for unoriented knots. I have exactly one R matrix that suggests itself, but it only works for oriented knots.)