Fusion of Jones-Wenzl colors

In the reduced theory the product of colors a and b decomposes with multiplicity one into colors c satisfying $|a-b|\le c\le \min (a+b,2r-4-a-b)$ and $a+b+c$ even. These rules follow by iterating $1\otimes j=(j-1)\oplus (j+1)$, with color $r-1$ set to zero. The closure traces give $d_a{d}_b={\sum }_c{N}_{ab}^c{d}_c$, and self-duality makes the multiplicities symmetric in all three colors.

In the reduced Jones-Wenzl theory, bend a fixed self-dual color a in the mutually inverse inclusion/projection maps of $c\subset b\otimes a$, obtaining $U_c^b:b\to c\otimes a$ and $V_c^b:c\otimes a\to b$. Cyclic closure trace and straightening cups and caps give $V_c^b{U}_c^b=(d_c/d_b)1_b$. Completeness of the normalized maps for all b then proves the weighted resolution. The dimensions are nonzero in the reduced theory. This local identity gives the precise weight preservation in the Kirby color handle-slide identity.

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.