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**Invariants of 3-manifolds via link polynomials and quantum groups.**
*(English)*
Zbl 0725.57007

Using techniques of quantum field theory, Witten defined, on a physical level of rigor, a series of new invariants of 3-manifolds and of links in 3-manifolds. The present paper provides a mathematical background for the construction of these invariants. For closed 3-manifolds, they form a sequence of complex numbers parametrized by complex roots of unity which specialize to the values of the Jones polynomial at the corresponding roots of unity for a link in the 3-sphere. For manifolds with boundary they form a sequence of finite dimensional complex linear operators producing for each root of unity a 3-dimensional topological quantum field theory. The paper uses the algebraic language of Hopf algebras. A central ingredient of the construction of the new invariants are the Kirby-moves relating different surgery presentations by framed links in the 3-manifold. Certain invariants of framed links in the 3-sphere combine in expressions invariant under these Kirby-moves thus giving the new invariants. The basic invariant used is the Jones polynomial which is known to be connected with the quantum enveloping algebra of the Lie algebra \(sl_ 2({\mathbb{C}})\) and its fundamental representation; other irreducible representations of the algebra are used to construct more basic invariants of links in the 3-sphere, which serve then as a groundstock for the construction of the invariants of general 3- manifolds.

A purely topological and combinatorial construction of the invariants, thus avoiding the algebraic machinery of the present paper, can be found in papers by W. B. R. Lickorish [Pac. J. Math. 149, No.2, 337-347 (1991)] and K. H. Ko and L. Smolinsky [ibid., 319-336 (1991)]. Concrete computations or applications of the new invariants have not yet appeared.

A purely topological and combinatorial construction of the invariants, thus avoiding the algebraic machinery of the present paper, can be found in papers by W. B. R. Lickorish [Pac. J. Math. 149, No.2, 337-347 (1991)] and K. H. Ko and L. Smolinsky [ibid., 319-336 (1991)]. Concrete computations or applications of the new invariants have not yet appeared.

Reviewer: B.Zimmermann (Trieste)

### MSC:

57M99 | General low-dimensional topology |

81T99 | Quantum field theory; related classical field theories |

### Keywords:

Jones polynomial; link; 3-sphere; 3-dimensional topological quantum field theory; Kirby-moves; surgery; quantum enveloping algebra
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\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)

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