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Quantum invariants of knots and 3-manifolds. 2nd revised ed. (English) Zbl 1213.57002
de Gruyter Studies in Mathematics 18. Berlin: Walter de Gruyter (ISBN 978-3-11-022183-1/hbk; 978-3-11-022184-8/ebook). xii, 592 p. (2010).
In 1994, Vladimir Turaev published a monograph titled [Quantum Invariants of Knots and 3-Manifolds. de Gruyter Studies in Mathematics. 18. Berlin: Walter de Gruyter. (1994; Zbl 0812.57003)]. The goal of the book was to present a systematic introduction to the author’s work with Reshetikhin (Reshetikhin-Turaev invariants) and with Viro (Turaev-Viro invariants). En-route, the reader is treated to an excellent introduction to the topological aspects of quantum topology.
The book was an instant classic. A decade and a half later, beyond its original purpose as a research monograph, Turaev’s book has established itself as a fundamental textbook in quantum topology. Its treatment of modular categories, of modular functors, and of TQFT has stood the test of time and in many ways is still unsurpassed.
Due to the strong appeal and wide use of this monograph, de Gruyter have published a second revised edition. It is essentially identical to the first edition, save some minor corrections and additions, and the deletion of an outdated list of research problems.
In the reviewer’s opinion, this book continues to be essential reading for anyone who wants to be a quantum topologist.

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