Hopf pairing of Jones-Wenzl colors ($H_{ij}=(-1{)}^{i+j}[(i+1)(j+1)]$)

The full bracket of a zero-framed Hopf link colored i and j is the displayed pairing. Its normalized matrix is the finite sine transform with parity signs. Sine orthogonality gives $H^2={\mathcal{D}}^{2}I$, where ${\mathcal{D}}^2={\sum }_j{d}_j^2=r/(2{\sin }^2(\pi /r))$. Thus the pairing is nondegenerate in the reduced theory.