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Racah coefficients and extended HOMFLY polynomials for all 5-, 6- and 7-strand braids. (English) Zbl 1262.81073
Summary: 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.

81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
22E70 Applications of Lie groups to the sciences; explicit representations
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