In famous Witten's paper "Quantum Field Theory and Jones Polynomial", Witten proposed
$\int DA \exp{iL} \prod_{k=1}^{r} W_{R_t} (C_i)$
as the knot invariant in ARBITRARY 3-manifold. ($L$ is a Langlangian of Chern-Simons 3-form, and the integral is Feynman path integral of gauge transformation.)
After Witten's paper, Turaev formulated Witten's idia as the quantum invariants. But Quantum invariants are defined only for knots in $S^3$. Does anyone know "Quantum invariants in arbitrary 3-manifold"?
Cf 1.Witten's paper and Turaev's paper https://projecteuclid.org/download/pdf_1/euclid.cmp/1104178138 http://mathlab.snu.ac.kr/~top/quantum/article/Reshetikhin01.pdf
2.knot theory in $RP^3$ https://arxiv.org/pdf/math/0312205.pdf
3.related question Unknown Witten-Reshetikhin-Turaev (WRT) Invariants for a 3 Manifold