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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.

MSC:
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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