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  "body": "\n## Summary\n\nQuantum invariants of $3$-manifolds are invariants of closed oriented [[3-manifolds|R20TB2KEQQ]].\nThey are built from the representation theory of quantum groups at roots of unity, or more generally from modular tensor categories.\nThe main examples are the Witten–Reshetikhin–Turaev invariants, which Witten predicted from quantum [[Chern–Simons theory|F28G886PWD]] and Reshetikhin and Turaev constructed rigorously.\nThey are computed from a surgery presentation of the manifold, using colored versions of the [[Jones polynomial|4SPC17FPXE]].\nThey are part of [[topological quantum field theories|DAP92B80J2]] in dimension $3$.\n\n## Explanations\n\n### L1 human\n\nEvery closed oriented $3$-manifold can be obtained by surgery on a link in the $3$-sphere, but in many different ways.\nA quantum invariant assigns a number to the link, built from knot polynomials, and corrects it so that the number does not depend on which link is chosen.\nThe result is a number attached to the manifold itself.\nThese numbers can be computed from a picture of the link, yet their geometric meaning is still only partly understood.\n",
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