## The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl$$(2,\mathbb{C})$$.(English)Zbl 0745.57006

Using the theory of quantum groups, Reshetikhin and Turaev defined invariants, associated with a simple Lie algebra, for a framed link $$L$$ in $$S^ 3$$. When the algebra is sl$$(2,\mathbb{C})$$ the values of these invariants at a fixed $$r$$th root of unity may be combined to produce a complex-valued invariant of the oriented 3-manifold obtained by surgery on $$L$$. The authors give a self-contained proof of the existence of these 3-manifold invariants for sl$$(2,\mathbb{C})$$ and the root $$e^{2\pi i/r}$$. The invariant, $$\tau_ r(M)$$, is a linear combination of coloured (by $${\mathbf k}$$) framed link invariants $$J_{L,{\mathbf k}}$$ of a framed link associated with $$M$$, where the $$J_{L,{\mathbf k}}$$ generalize the Jones polynomial. A cabling formula reducing $$J_{L,{\mathbf k}}$$ to the Jones polynomial of cables of $$L$$ is given. A symmetry principle which is established has many consequences: it reduces the number of steps required in calculating $$\tau_ r(M)$$; for odd $$r$$ it enables $$\tau_ r(M)$$ to be expressed as the product of $$\tau_ 3(M)$$ or $$\overline{\tau_ 3(M)}$$ and another invariant $$\tau_ r'(M)$$; for even $$r$$ it enables $$\tau_ r(M)$$ to be split into a sum of invariants of $$M$$ with extra structure; it links the mod 5 Casson invariant of a homology sphere obtained by Dehn surgery on a knot to $$\tau_ 5(M)$$, etc.

### MSC:

 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M25 Knots and links in the $$3$$-sphere (MSC2010) 17B37 Quantum groups (quantized enveloping algebras) and related deformations 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory

Mathematica
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