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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a modular tensor category.\nAn 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.\nThe 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 $\\mathcal{C}$.\n\nThe *quantum invariant* $\\tau_{\\mathcal{C}}(M)$ of a closed oriented [[3-manifold|R20TB2KEQQ]] $M$ is obtained as follows.\n\n1. Present $M$ by integral [[Dehn surgery|ZA405B0Z1S]] on a framed link $L \\subset S^3$ with $n$ components.\n2. Color each component of $L$ by the Kirby color $\\omega = \\sum_i \\dim(V_i)\\, V_i$, and evaluate $\\langle L \\rangle$ by linearity.\n   Here $\\dim$ is the quantum dimension, and the sum runs over the isomorphism classes of simple objects $V_i$.\n3. Multiply by a normalising factor that depends only on $\\mathcal{C}$, on $n$ and on the signature of the linking matrix of $L$.\n\nThe normalising factor is chosen so that the result does not change under the [[Kirby moves|AVQ9WN1MDV]], and hence depends only on $M$.\n\n## Remarks\n\nLet $\\mathcal{C}$ come from $U_q(\\mathfrak{sl}_2)$ with $q = e^{2\\pi i / r}$.\nThen the evaluation in step 2 is a combination of colored [[Jones polynomials|4SPC17FPXE]] of $L$, and $\\tau_{\\mathcal{C}}$ is the $SU(2)$ Witten–Reshetikhin–Turaev invariant at level $r - 2$.\nSuitably normalised, the invariant is multiplicative under connected sum.\nStep 3 is needed: the links $\\emptyset$ and the $(\\pm 1)$-framed unknot both present $S^3$, while their colored evaluations differ in general.\nNormalisation conventions differ between authors.\n",
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