The irreducible representations of two intermediate Casimir elements associated to the recoupling of three identical irreducible representations of U q ( s l 2 ) U_q(\mathfrak {sl}_2) are considered. It is shown that these intermediate Casimirs are related by a conjugation involving braid group representations. Consequently, the entries of the braid group matrices are explicitly given in terms of the q q -Racah polynomials which appear as 6 j 6j -symbols in the Racah problem for U q ( s l 2 ) U_q(\mathfrak {sl}_2) . Formulas for these polynomials are derived from the algebraic relations satisfied by the braid group representations.