In classical $\operatorname{SL}_2(\mathbb{C})$-Chern-Simons theory, the expression $\rho : \pi_1(M) \to \operatorname{SL}_2( \mathbb{C}) $ its complex volume $\operatorname{V}(M, \rho) \in \mathbb{C} / 2 \pi^2 i \mathbb{Z}$, where the real part is the volume and the imaginary part is the Chern-Simons invariant part. Existing literature focuses on computing $\operatorname{V}$ using triangulation. In this paper, we show how to calculate $\operatorname{V}(M, L, \rho)$ directly from the surgical diagram and the expression $\rho : \pi_1(M \setminus L) \to \operatorname{ SL}_2(\mathbb{C})$. If $M$ has non-empty bounds, then $\operatorname{V}(M, L, \rho)(\mathfrak{s})$ has some extra data $\mathfrak{s}$. Our method describes $\rho$ in a coordinate system closely related to the quantum group, and connects our construction to Witten-Reshetikhin-Turaev’s quantum $\operatorname{SU}(2)$ Chern-Simons theory. Think of it as the classic, non-compact version of .
