Hopf pairing of Jones-Wenzl colors
= Hopf pairing of Jones-Wenzl colors
{c}
{title2=$H_{ij}=(-1)^{i+j}[(i+1)(j+1)]$}
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 $H^2=\mathcal D^2I$, where $\mathcal D^2=\sum_jd_j^2=r/(2\sin^2(\pi/r))$. Thus the pairing is nondegenerate in the reduced theory.