## 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

### Citations:

Zbl 0779.57009; Zbl 0725.57007; Zbl 0667.57005
Full Text:

### References:

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