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The net chiral central charge of an edge theory. For a bosonic unitary modular tensor category, its residue modulo eight is determined by the topological spins and quantum dimensions through $e^{2\pi ic/8}={\mathcal{D}}^{-1}{\sum }_a{d}_a^2{\theta }_a$. This phase corrects the signature dependence in the Witten-Reshetikhin-Turaev invariant.

The symmetric matrix with the component framing integers on the diagonal and the pairwise linking numbers off the diagonal. Its signature enters the Witten-Reshetikhin-Turaev invariant.

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.