Racah coefficients and extended HOMFLY polynomials for all 5-, 6- and 7-strand braids

Basing on evaluation of the Racah coefficients for SU_q(3) (which supported the earlier conjecture of their universal form) we derive explicit formulas for all the 5-, 6- and 7-strand Wilson averages in the fundamental representation of arbitrary SU(N) group (the HOMFLY polynomials). As an application, we list the answers for all 5-strand knots with 9 crossings. In fact, the 7-strand formulas are sufficient to reproduce all the HOMFLY polynomials from the katlas.org: they are all described at once by a simple explicit formula with a very transparent structure. Moreover, would the formulas for the relevant SU_q(3) Racah coefficients remain true for all other quantum groups, the paper provides a complete description of the fundamental HOMFLY polynomials for all braids with any number of strands.Comment: 16 pages + Tables and Appendice

Colored HOMFLY Polynomials as Multiple Sums over Paths or Standard Young Tableaux

If a knot is represented by an m-strand braid, then HOMFLY polynomial in representation R is a sum over characters in all representations Q∈R⊗m. Coefficients in this sum are traces of products of quantum ℛ^-matrices along the braid, but these matrices act in the space of intertwiners, and their size is equal to the multiplicity MRQ of Q in R⊗m. If R is the fundamental representation R=[1]=□, then M□Q is equal to the number of paths in representation graph, which lead from the fundamental vertex □ to the vertex Q. In the basis of paths the entries of the m-1 relevant ℛ^-matrices are associated with the pairs of paths and are nonvanishing only when the two paths either coincide or differ by at most one vertex, as a corollary ℛ^-matrices consist of just 1×1 and 2×2 blocks, given by very simple explicit expressions. If cabling method is used to color the knot with the representation R, then the braid has as many as m|R| strands; Q have a bigger size m|R|, but only paths passing through the vertex R are included into the sums over paths which define the products and traces of the m|R|-1 relevant ℛ^-matrices. In the case of SU(N), this path sum formula can also be interpreted as a multiple sum over the standard Young tableaux. By now it provides the most effective way for evaluation of the colored HOMFLY polynomials, conventional or extended, for arbitrary braids