Kirby killing property ($K_{\Omega }(x)=\mathcal{D}{\delta }_{x1}$)

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.