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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

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