Colored link

A colored link assigns each component a color. In the trivial coloring all components have the same color, so a doubly slice realization uses one unknotted surface component.

The Kirby color is the weighted sum of annular closures of the surviving Jones-Wenzl idempotents at a root of unity. Its weights are their closed bracket traces. The fusion of Jones-Wenzl colors gives $a\Omega =d_a\Omega$ in the annular fusion ring. Inserting the corresponding fusion resolutions along a band gives the Kirby color handle-slide identity, the central ingredient of surgery invariance.

The Kirby color divided by the total quantum dimension. A zero-framed closed loop with this color evaluates to $\mathcal{D}$ in the isotopy normalization of anyon diagrams. An encircling loop acts on a simple charge $x$ by $\mathcal{D}{\delta }_{x1}$, the Kirby killing property. Normalization must be tracked explicitly when using a surgery formula.

In a modular tensor category, a loop with the normalized Kirby color encircling a charge-$x$ strand projects onto the vacuum anyon charge and multiplies it by $\mathcal{D}$. Indeed, the encircling scalar is ${\mathcal{D}}^{-1}{\sum }_a{d}_a{S}_{ax}/S_{1x}$, which equals $\mathcal{D}{\delta }_{x1}$ by unitary matrix orthogonality for the modular S-matrix.

An encircling normalized Kirby color loop around two equally oriented anyon strands is $\mathcal{D}$ times the orthogonal projection onto their total vacuum anyon charge. In the isotopy normalization of anyon diagrams, erasing the vacuum edge writes this as ${\delta }_{b,\bar{a}}(\mathcal{D}/d_a)$ times the cup-cap diagram. The cup-cap squared is $d_a$ times the cup-cap, so $d_a^{-1}$ is essential.

For a charge-$a$ component threading a pair of zero-framed auxiliary components in the standard Borromean rings surgery arrangement, color both auxiliary components by the normalized Kirby color. Relative to the bare $a$ strand, eliminating this pair contributes $H_a={\mathcal{D}}^2/d_a^2$. The two-strand Kirby projector applied to the two oppositely oriented portions of the $a$ component supplies a factor $\mathcal{D}/d_a$ at each of the two eliminations. This local evaluation underlies the surgery calculation of the handle-cut formula for anyon ground-state degeneracy.

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.

The bracket of a framed link with every surgery component colored by the Kirby color is unchanged by a handle slide. For a fixed self-dual color a choose inclusion/projection maps $i_c:c\to b\otimes a$, $p_c:b\otimes a\to c$ with $p_c{i}_c=1_c$ and ${\sum }_c{i}_c{p}_c=1_{b\otimes a}$. Bend the a strand to form $U_c^b:b\to c\otimes a$ and $V_c^b:c\otimes a\to b$. Closing the diagrams gives $V_c^b{U}_c^b=(d_c/d_b)1_b$. Completeness of these dual fusion maps therefore gives ${\sum }_b{d}_b{U}_c^b{V}_c^b=d_c{1}_{c\otimes a}$. Resolve the strand meeting a slide band into these channels. Moving the band across the colored loop bends its maps into U and V, and this weighted identity replaces the old loop-color sum by precisely the new Kirby coefficient $d_c$. The exterior ribbon diagrams agree by isotopy. Summing the sliding color with its own Kirby weight proves the full identity, including its carried band framing.