Quantum invariant of 3-manifolds

Definition

Let $\mathcal{C}$ be a modular tensor category. An example is the category of tilting representations of the quantum group $U_q(\mathfrak{sl}_2)$ at a root of unity $q$, modulo negligible morphisms. The Reshetikhin–Turaev functor of $\mathcal{C}$ assigns a number $\langle L \rangle$ to each framed link $L$ in $S^3$ whose components are colored by objects of $\mathcal{C}$.

The quantum invariant $\tau_{\mathcal{C}}(M)$ of a closed oriented 3-manifold $M$ is obtained as follows.

1. Present $M$ by integral Dehn surgery on a framed link $L \subset S^3$ with $n$ components.
2. Color each component of $L$ by the Kirby color $\omega = \sum_i \dim(V_i)\, V_i$, and evaluate $\langle L \rangle$ by linearity. Here $\dim$ is the quantum dimension, and the sum runs over the isomorphism classes of simple objects $V_i$.
3. Multiply by a normalising factor that depends only on $\mathcal{C}$, on $n$ and on the signature of the linking matrix of $L$.

The normalising factor is chosen so that the result does not change under the Kirby moves, and hence depends only on $M$.