Ribbon graphs and their invariants derived from quantum groups.

*(English)*Zbl 0768.57003The authors construct the generalization of the Jones polynomial of links to the case of graphs in \(R^ 3\). They introduce the so-called coloured ribbon graphs in \(R^ 2 \times [0,1]\) and define for them Jones-type isotopy invariants. The approach to colouring is based on Drinfeld’s notion of a quasitriangular Hopf algebra. For each quasitriangular Hopf algebra \(A\) the authors define \(A\)-coloured (ribbon) graphs. The colour of an edge is an \(A\)-module. The colour of a vertex is an \(A\)-linear homomorphism intertwining the modules which correspond to edges incident to this vertex. The category of an \(A\)-coloured ribbon graph is a compact braided strict monoidal category introduced by A. Joyal and R. Street [Braided monoidal categories, Macquarie Math. Reports, Report No. 860081 (1986)]. If \(A\) satisfies a minor additional condition, then the authors construct a canonical covariant functor from the category of \(A\)- coloured ribbon graphs into the category of \(A\)-modules. In the case of \(A\) being the quantized universal enveloping algebra of \(sl_ 2\) this functor generalizes the Jones polynomial of links. If \(A=U_ h(sl_ n(C))\) this generalizes the Jones-Conway (Thomflyp) polynomial and for \(A=U_ hG\), \(G=so(n)\), \(sp(2k)\) the Kauffman polynomial. The paper under review was followed by the authors’ paper “Invariants of 3-manifolds via link polynomials and quantum groups” [Invent. Math. 103, 547-597 (1991; Zbl 0725.57007)]in which the authors construct new topological invariants of compact oriented manifolds (employing the methods of the paper under review). The construction was partially inspired by ideas of E. Witten [Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121, 351-399 (1989; Zbl 0667.57005)]who considered quantum field theory defined by the nonabelian Chern-Simon action and applied it to the study of 3-manifolds (on physical level of rigor).

Reviewer: J.H.Przytycki (Riverside)

##### MSC:

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |

##### Keywords:

generalization of the Jones polynomial; graphs in \(R^ 3\); coloured ribbon graphs; quasitriangular Hopf algebra; quantized universal enveloping algebra of \(\text{sl}_ 2\); Kauffman polynomial; quantum field theory
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\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)

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