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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

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    $\begingroup$ I just came across this ancient question. For future readers, I note that in principle calculating colored HOMFLY polynomials always reduces to calculating the usual HOMFLY polynomial of some linear combination of knots. The reason is that HOMFLY of (K colored by a given representation) is equal to HOMFLY of (K satellited by the corresponding Aiston-Morton idempotent). There are explicit formulas for the latter as linear combinations of links in the annulus. $\endgroup$ Feb 1 '18 at 17:42
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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.)

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Recently, there has been some progress on quantum $6j$-symbols for $U_q(\mathfrak{sl}_N)$: Nawata-Ramadevi-Zodinmawia for symmetric representations, and Gu-Jockers for the representation specified by the (2,1) Young diagram. In particular, the latter one deals with quantum $6j$-symbols with multiplicity structure.

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