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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 $U_q({\mathfrak{sl}_n})$.

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

Is 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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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 with (a few horizontal boxes of Young TableauxTableaux)?

I do not know how one can calculate colored HOMFLY-PT polynomials for non-torus knots without information about quantum $6j$-symbols for $U_q({\mathfrak{sl}_n})$.

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

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).

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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$6j$-symbols for $U_q({\mathfrak{sl}}_n)$ and colored HOMFLY polynomials

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 symmetric representation with a few 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 $U_q({\mathfrak{sl}_n})$.

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

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).

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