Gauss sums of Jones-Wenzl colors ($p_{\pm }={\sum }_j{d}_j^2{\theta }_j^{\pm 1}$)

These sums are the evaluations of the plus- and minus-one-framed unknots with Kirby color. Twist balancing and the fusion-dimension identity give $H({\theta }_j{d}_j{)}_j=p_{+}({\theta }_j^{-1}{d}_j{)}_j$. Conjugating this identity and using $H^2={\mathcal{D}}^{2}I$ proves $p_{+}{p}_{-}={\mathcal{D}}^2$, so neither stabilization factor vanishes.

Let $H$ be the real Hopf pairing of Jones-Wenzl colors matrix, $d$ the dimension vector and $T$ the diagonal matrix of Jones-Wenzl twist eigenvalues. Twist balancing on the fusion of Jones-Wenzl colors gives $HTd=p_{+}{T}^{-1}d$, using ${\sum }_i{d}_i{N}_{ij}^k=d_j{d}_k$. Complex conjugation gives $H{T}^{-1}d=p_{-}Td$. Apply $H$ again and use finite sine orthogonality, $H^2={\mathcal{D}}^{2}I$. Since $Td\ne 0$, the displayed identity follows. Thus both factors used to normalize the Jones polynomial surgery invariant are nonzero.