Quantum invariants of knots and 3-manifolds

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This monograph is now available in its second revised edition due to its strong appeal and wide use. It systematically explores 3-dimensional topological quantum field theories (TQFTs), building on the author's collaboration with N. Reshetikhin and O. Viro. The subject is rooted in the discovery of the Jones polynomial of knots and Witten-Chern-Simons field theory. The algebraic study of 3-dimensional TQFTs is influenced by braided categories and quantum groups. The book is divided into three parts: Part I constructs 3-dimensional TQFTs and 2-dimensional modular functors from modular categories, yielding a broad range of knot and 3-manifold invariants, along with linear representations of surface mapping class groups. Part II employs 6j-symbols to define state sum invariants of 3-manifolds, establishing their connection to the TQFTs from Part I through the theory of shadows. Part III presents constructions of modular categories derived from quantum groups and skein modules of tangles in 3-space. This essential contribution to topological quantum field theory is accessible to graduate students in mathematics and physics with a background in basic algebra and topology, serving as a vital resource for those eager to explore this intriguing intersection of mathematics and physics.