Question

Explicit expression of quantum $6j$-symbolos for $U_q({\mathfrak{sl}_2})$ have been known due to the work of Kirillov and Reshitikhin.

My Question:

How much are known about quantum $6j$-symbolos for $U_q({\mathfrak{sl}_n})$? Are there partial results, say when the highest weights are given by small rank symmetric representation (a few horizontal boxes of Young Tableaux)?

I do not know how one can calculate colored HOMFLY-PT polynomials for non-torus knots without information about quantum $6j$-symbols for $U_q({\mathfrak{sl}_n})$. Physicists guessed the form of HOMFLY-PT polynomials colored by symmetric representation for the figure-eight (See the paper by the ITEP group and the recent paper). However, I want to know more examples for colored HOMFLY-PT polynomials.

Are there explicit formulae of colored HOMFLY-PT polynomials of non-torus knots?

In addition,

is there any way to calculate colored HOMFLY-PT polynomials of non-torus knots without using quantum $6j$-symbolos for $U_q({\mathfrak{sl}_n})$?

A similar question can be found here

Answer 1

Provided you have some "base data" (closed formulae for quadratic Casimirs and quantum dimensions, for instance), for "low" irreps the quantum 6j symbols might be calculated recursively using Biedenharn-Elliott and other equations (e.g. those following from Reidemeister for 3-nodes). At the moment I'm exactly doing this for the whole E7 series, which is even worse in complexity, and see no principal obstacles - Surgeons Warning: It's extremely time-consuming due to combinatoric explosion. Also note that as soon as a multiplicity>1 pops up (the irrep 11 of A2 coming to mind immediately) you are in even more trouble.
(I'm no expert and always rely on brute force calculations - for a more math oriented approach I'm no help at all.)