I always wondered why one writhe unit (read: colored with the irrep of a quantum Lie algebra and evaluated as Reshetikhin-Turaev invariant thereof) is essentially the quadratic casimir of that irrep. (With a proper normalization. Since quadratic casimirs are defined in a complete other way via traces over tensors, I don't know if that's proven at all, and frankly I don't even know whether quantum Lie algebras have casimirs the same way as their "normal" cousins.)
As usual, after a wild generalization everything fell in place:
- A knot without any intersection is a circle (duh) and its Reshetikhin-Turaev invariant is the quantum dimension, i.e. the 0th casimir.
- A knot with one intersection (see above), 2nd casimir (times 0th for the loop itself).
- A Hopf link (colored with same irrep) with two intersections I thus would expect to be expressible with the 4th, 2nd and 0th casimir.
- Trefoil with 6th...and so on.
This wild hypothesis even would give some sense to why the "easy" (skein-relation wise) Lie algebras are those with the smallest ranks: A1, G2, B2.
So is my hypothesis true - the Reshetikhin-Turaev invariant of a knot/link with n crossings colored with an irrep can be expressed in maximally the 2n-th casimirs (maybe also less) of this irrep? Possible even as linear combinations (in the sense e.g. $a*C_4+b*C_2^2+c*C_2$, so you can say the value of the Hopf link is a quartic Casimir, times dimension...i.e. a generalized 4th Dynkin index)?