The irreducible representations of two intermediate Casimir elements associated to the recoupling of three identical irreducible representations of U q (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-Racah polynomials which appear as 6j-symbols in the Racah problem for U q (sl 2). Formulas for these polynomials are derived from the algebraic relations satisfied by the braid group representations.