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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.

Let $B_c^{ab}$ denote the split-fuse diagram with a charge-$c$ edge joining the lower fusion vertex to the upper splitting vertex, and outer strands $a,b$. Completeness in the isotopy normalization of anyon diagrams readsResolve the two strands inside the encircling normalized Kirby color loop in this way. Move that loop onto the intermediate charge-$c$ edge. The Kirby killing property multiplies it by $\mathcal{D}{\delta }_{c1}$, leavingThe vacuum anyon channel is allowed only for $b=\bar{a}$, and then $d_b=d_a$ and $d_1=1$. Erasing the vacuum edge turns $B_1^{a\bar{a}}$ into the cup-cap diagram $E_a$, which connects the incoming pair by a cap and the outgoing pair by a cup. Thus the requested two-strand Kirby projector isHere $P_1^{a\bar{a}}=E_a/d_a$ is the orthogonal projection onto total vacuum anyon charge; for $b\ne \bar{a}$ it is zero. Both strands in the source point upwards, so the condition is $b=\bar{a}$, not $b=a$ unless $a$ is self-dual.

The coefficient can be checked without a convention guess: stacking two cup-cap diagrams creates one charge-$a$ loop, so $E_a^2=d_a{E}_a$. Consequently $(E_a/d_a{)}^2=E_a/d_a$, as required of an orthogonal projection. Omitting $1/d_a$ would fail this check.

Use the Witten-Reshetikhin-Turaev invariant with the source's normalized Kirby color on all $2g+1$ surgery components. Since the signature of the linking matrix is zero, the framing-anomaly factor is one, and the TQFT partition function isAll evaluations in this part use the isotopy normalization of anyon diagrams and preserve the indicated blackboard framing.

Expand the long component as ${\mathcal{D}}^{-1}{\sum }_a{d}_{a}a$ and fix its charge $a$. Each pair of auxiliary components is in the Borromean rings arrangement with this component. Eliminate such a pair using the two-strand Kirby projector from c: bend the long component to expose its two oppositely oriented portions, viewed as equally oriented strands labeled $a,\bar{a}$. The first encircling normalized Kirby color loop replaces these by a vacuum cup-cap with coefficient $\mathcal{D}/d_a$. This vacuum connection opens the second encircling loop so that the same two-strand Kirby projector applies once more. Removing the resulting vacuum edges and straightening the bends restores the original charge-$a$ strand. Thus the Borromean handle-pair evaluation multiplies the bare strand byThere is no residual twist. This is a relative strand evaluation: the final closure of the long component is counted separately. As a normalization check, when $a=1$ the two auxiliary components become two unlinked normalized Kirby color loops, each of value $\mathcal{D}$, giving $H_1={\mathcal{D}}^2$.

The $g$ pairs occupy disjoint portions of the long component, so their factors multiply. Closing that component gives its quantum dimension $d_a$. Combining this with its normalized Kirby color coefficient $d_a/\mathcal{D}$ givesApplying the overall surgery prefactor yieldsFinally, the topological quantum field theory assigns the identity map to a cylinder over ${\Sigma }_g$. Closing its time direction takes the trace of that identity, soThis independently reproduces b. For $g=0$ the surgery link is one zero-framed normalized Kirby color unknot: its value $\mathcal{D}$ cancels the surgery prefactor and gives $Z(S^2\times S^1)=1$. For $g=1$ the result is the number of simple anyons, as required for the torus.