Jones-Wenzl idempotent ($f_n$)

The Jones-Wenzl idempotent has identity-diagram coefficient one, satisfies $f_n^2=f_n$, and annihilates each adjacent cap-cup generator on either side. With $\delta =-A^2-A^{-2}$ and ${\Delta }_0=1$, ${\Delta }_1=\delta$, ${\Delta }_n=\delta {\Delta }_{n-1}-{\Delta }_{n-2}$, it is constructed recursively by $f_n=F-({\Delta }_{n-2}/{\Delta }_{n-1})F{e}_{n-1}F$, where $F=f_{n-1}\otimes 1$. The required denominators must be nonzero. Its closed bracket trace is ${\Delta }_n=(-1{)}^n[n+1]$.

Close the last strand of the projector recursion. Closing $f_{n-1}\otimes 1$ gives $\delta f_{n-1}$, while closing its cap-cup correction gives $f_{n-1}$. The coefficient is $\delta -{\Delta }_{n-2}/{\Delta }_{n-1}={\Delta }_n/{\Delta }_{n-1}$. Iteration proves $\mathrm{tr}(f_n)={\Delta }_n$ and supplies the normalization needed to split the two fusion of Jones-Wenzl colors channels in $f_{n-1}\otimes 1$.

At $A=e^{\pi i/(2r)}$, $r\ge 3$, the projector $f_{r-1}$ exists but its closed trace is zero. Quotient by the skein ideal generated by this projector. The surviving simple colors are $0,\dots ,r-2$, with nonzero quantum dimensions $d_j=(-1{)}^j\sin ((j+1)\pi /r)/\sin (\pi /r)$. This truncation prevents zero-trace channels from interfering with nondegenerate fusion and surgery normalization.

A surviving color denotes $j$ parallel framed strands equipped with the Jones-Wenzl idempotent $f_j$. Its closed trace is its quantum dimension. The semisimplicity of the reduced Temperley-Lieb category identifies these colors with the simple objects; their tensor products decompose according to fusion of Jones-Wenzl colors.

The Jones-Wenzl idempotent recursion decomposes $f_j\otimes 1$ into the orthogonal $f_{j+1}$ and $f_{j-1}$ channels. The Jones-Wenzl partial trace supplies the nonzero scalar normalizing the cap-cup channel. At the highest surviving color the $f_{r-1}$ channel is zero. Induction therefore decomposes every tensor power into the surviving colors. A planar matching between different projected colors has a turnback on the larger side and vanishes; between equal colors only the identity matching remains. Their endomorphism spaces are consequently one-dimensional, and the resulting finite matrix blocks split. Taking direct sums and splitting these idempotents produces a semisimple category, supplying the complete resolutions needed in fusion of Jones-Wenzl colors and the Kirby color handle-slide identity.

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.

A positive framing twist on an n-strand Jones-Wenzl cable has n positive individual curls and $n(n-1)$ braid crossings. Each curl contributes $-A^3$, and each braid crossing contributes $A$ because its turnback term annihilates the Jones-Wenzl idempotent. Their product is the displayed scalar. This is the colored framing factor used in surgery invariants.