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Basing on evaluation of the Racah coefficients for SU_q(3) (which supportedthe earlier conjecture of their universal form) we derive explicit formulas forall the 5-, 6- and 7-strand Wilson averages in the fundamental representationof arbitrary SU(N) group (the HOMFLY polynomials). As an application, we listthe answers for all 5-strand knots with 9 crossings. In fact, the 7-strandformulas are sufficient to reproduce all the HOMFLY polynomials from thekatlas.org: they are all described at once by a simple explicit formula with avery transparent structure. Moreover, would the formulas for the relevantSU_q(3) Racah coefficients remain true for all other quantum groups, the paperprovides a complete description of the fundamental HOMFLY polynomials for allbraids with any number of strands.

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