## Evaluations of the 3-manifolds invariants of Witten and Reshetikhin- Turaev for $$sl(2,C)$$.(English)Zbl 0756.57006

Geometry of low-dimensional manifolds. 2: Symplectic manifolds and Jones- Witten-Theory, Proc. Symp., Durham/UK 1989, Lond. Math. Soc. Lect. Note Ser. 151, 101-114 (1990).
[For the entire collection see Zbl 0722.00024.]
Fix an integer $$r>1$$. The 3-manifold invariant $$\tau_ r$$ assigns a complex number $$\tau_ r(M)$$ to each oriented, closed, connected 3- manifold $$M$$ and satisfies: (1) $$\tau_ r(M\#N)=\tau_ r(M)\cdot\tau_ r(N)$$; (2) $$\tau_ r(-M)=\overline{\tau_ r(M)}$$; (3) $$\tau_ r(S^ 3)=1$$.
$$\tau_ r(M)$$ is defined as a weighted average of colored, framed link invariants $$J_{L,k}$$ of a framed link $$L$$ for $$M$$, where a coloring of $$L$$ is an assignment of integers $$k_ i$$, $$0<k_ i<r$$, to the components $$L_ i$$ of $$L$$. The $$k_ i$$ denote representations of the Hopf algebra of dimension $$k_ i$$, and $$J_{L,k}$$ is a generalization of the Jones polynomial of $$L$$ at $$q$$. The authors describe the Reshetikhin-Turaev version of $$\tau_ r$$ for $$q=e^{2\pi i/r}$$, giving a cabling formula, a symmetry principle, and evaluations at $$r=3,4$$ and $$6$$.
Reviewer: A.K.Guts (Omsk)

### MSC:

 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M99 General low-dimensional topology 82B23 Exactly solvable models; Bethe ansatz 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory

Zbl 0722.00024