##
**Quantum invariants of knots and 3-manifolds.**
*(English)*
Zbl 0812.57003

de Gruyter Studies in Mathematics. 18. Berlin: Walter de Gruyter. x, 588 p. (1994).

The staggering array of relatively recent combinatorial-algebraic- topological methods of defining invariants of low-dimensional objects such as links, braids, 3-manifolds etc., is attacked in this big book. Utilizing the basic notion of a category of modules (modular category) equipped with a tensor product and various morphisms, depending on the need, geometric objects are given algebraic realizations. Algebraic invariants can then be associated with topological spaces.

The book contains many ideas and the more or less random sampling from each chapter given below may give some idea of the contents: Chapter I. Invariants of graphs in Euclidean 3-space: 1. Ribbon categories. Chapter II. Invariants of closed 3-manifolds: 2. Modular tensor categories; 5. Hermitian and unitary categories. Chapter III. Foundations of topological quantum field theory: 1. Axiomatic definition of TQFT’s; 4. Quantum invariants; 5. Hermitian and unitary TQFT’s. Chapter IV. Three- dimensional topological quantum field theory: 3. Lagrangian relations and Maslov indices; 5. Action of the modular groupoid; 9. Anomaly-free TQFT; 10. Hermitian TQFT; 11. Unitary TQFT; 12. Verlinde algebra. Chapter V. Two-dimensional modular functors: 3. Weak and mirror modular functors. Chapter VI. \(6j\)-symbols: 3. Symmetrized multiplicity modules; 4. Framed graphs. Chapter VII. Simplicial state sums on 3-manifolds: 3. Simplicial 3-dimensional TQFT. Chapter VIII. Generalities on shadows: 3. Shadow links; 4. Surgeries on shadows; 7. Shadow graphs. Chapter IX. Shadows of manifolds: 1. Shadows of 4-manifolds; 8. Shadows of framed graphs. Chapter X. State sums on shadows: 1. State sum models on shadowed polyhedra; 7. Invariants of framed graphs from the shadow viewpoint; 9. computations for graph manifolds. Chapter XI. An algebraic construction of modular categories: 1. Hopf algebras and categories of representations; 6. Quantum groups at roots of unity. Chapter XII. A geometric construction of modular categories: 1. Skein modules and the Jones polynomial; 3. The Temperley-Lieb algebra; 4. The Jones-Wenzl idempotents; 8. Multiplicity modules. Appendix I. Dimension and trace re- examined. Appendix II. Vertex models on link diagrams. Appendix III. Gluing re-examined. Appendix IV. The signature of closed 4-manifolds from a state sum.

The book contains many ideas and the more or less random sampling from each chapter given below may give some idea of the contents: Chapter I. Invariants of graphs in Euclidean 3-space: 1. Ribbon categories. Chapter II. Invariants of closed 3-manifolds: 2. Modular tensor categories; 5. Hermitian and unitary categories. Chapter III. Foundations of topological quantum field theory: 1. Axiomatic definition of TQFT’s; 4. Quantum invariants; 5. Hermitian and unitary TQFT’s. Chapter IV. Three- dimensional topological quantum field theory: 3. Lagrangian relations and Maslov indices; 5. Action of the modular groupoid; 9. Anomaly-free TQFT; 10. Hermitian TQFT; 11. Unitary TQFT; 12. Verlinde algebra. Chapter V. Two-dimensional modular functors: 3. Weak and mirror modular functors. Chapter VI. \(6j\)-symbols: 3. Symmetrized multiplicity modules; 4. Framed graphs. Chapter VII. Simplicial state sums on 3-manifolds: 3. Simplicial 3-dimensional TQFT. Chapter VIII. Generalities on shadows: 3. Shadow links; 4. Surgeries on shadows; 7. Shadow graphs. Chapter IX. Shadows of manifolds: 1. Shadows of 4-manifolds; 8. Shadows of framed graphs. Chapter X. State sums on shadows: 1. State sum models on shadowed polyhedra; 7. Invariants of framed graphs from the shadow viewpoint; 9. computations for graph manifolds. Chapter XI. An algebraic construction of modular categories: 1. Hopf algebras and categories of representations; 6. Quantum groups at roots of unity. Chapter XII. A geometric construction of modular categories: 1. Skein modules and the Jones polynomial; 3. The Temperley-Lieb algebra; 4. The Jones-Wenzl idempotents; 8. Multiplicity modules. Appendix I. Dimension and trace re- examined. Appendix II. Vertex models on link diagrams. Appendix III. Gluing re-examined. Appendix IV. The signature of closed 4-manifolds from a state sum.

Reviewer: L.P.Neuwirth (Princeton)

### MSC:

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

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

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |

18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |

18D20 | Enriched categories (over closed or monoidal categories) |

18D35 | Structured objects in a category (MSC2010) |

57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |

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