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Quantum invariants of links and 3-valent graphs in 3-manifolds. (English) Zbl 0822.57012

The author gives algebraic and combinatorial definitions of the SU(2) quantum invariants of a pair \((M,\Gamma)\) with \(M\) a compact 3-manifold and \(\Gamma\) an embedded 3-valent fat graph (graph with framing) colored with irreducible representations of SU(2). To define these invariants he uses a triangulation of \((M,\Gamma)\) and associates each simplex with a quantum \(6j\)-symbol.
These invariants are generalizations of the Turaev-Viro invariants [the author and O. Y. Viro, Topology 31, No. 4, 865-902 (1992; Zbl 0779.57009)], where the invariants are defined for the case that \(M\) is closed and \(\Gamma\) is empty. It was also proved by K. Walker and the author that these are the squares of the absolute values of the \(\text{sl}_ 2(\mathbb{C})\) invariants of N. Yu. Reshetikhin and the author [Invent. Math. 103, No. 3, 547-597 (1991; Zbl 0725.57007)]. Moreover it is shown in this paper that the invariants of \((S^ 3, L)\) are equal to the values of the framed version of the Jones polynomial (more precisely the Kauffman bracket) evaluated at roots of unity, where \(L\) is an unoriented framed link in \(S^ 3\) colored with the vector representation. This gives a mathematically rigorous 3-dimensional definition of the Jones polynomial. Note that a ‘physical’ definition using the Feynman path integral was given by E. Witten [Commun. Math. Phys. 121, 351-399 (1989; Zbl 0667.57005)] together with an intuitive definition of quantum invariants for more general compact Lie groups.
Reviewer: H.Murakami (Osaka)

MSC:

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

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