Quantum Invariants of Knots and 3-Manifolds by Turaev
summary
Here is a quick summary of its contents. The book begins by explaining the tangle and ribbon categories, and a graphical calculus for morphisms that allows movement from knots and links to algebra. The book then turns to the formulation of invariants of closed 3-manifolds via modular categories and surgery on links in the 3-sphere. This is a categorical formalization of link invariants obtained by labelling each link component with a (possibly different) representation of a quantum group. The book then (Chapter III) defines modular functors and an axiomatic definition of TQFTs. The next chapter examines the construction of 3-dimensional TQFTs based on invariants of 3-manifolds defined via surgery on links. Chapter 5 discusses the structure of 2-dimensional modular functors—an abstraction of key properties of 2-dimensional rational conformal field theory. Three-dimensional TQFTs can be described via 2-dimensional modular functors on marked surfaces. This ends Part I.
Part II is entitled "The Shadow World". The reference is to the paper by A. N. Kirillov and Reshetikhin 'in Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988), 285–339, World Sci. Publishing, Teaneck, NJ, 1989; MR1026957 (90m:17022)', where a diagrammatic approach to quantum 6j symbols leads to a startling reformulation of link invariants in terms of labellings of the 2-cell complex of the link diagram. Turaev carries this idea very far in Part II, giving flexible algebraic and geometric approaches to 6j symbols with applications to state sum models for 3-manifold invariants on triangulated 3-manifolds. Shadows then become a general theory that even encapsulates possible invariants of 4-manifolds, and generalizations of knots and links to a category of shlinks (shadow links). The theory of state sums on shadows yields a proof that the Turaev-Viro invariant (defined as a state sum on 3-manifold triangulations) is equal to the absolute square of the Witten-Reshetikhin-Turaev invariant (defined by surgery and quantum groups).
Part III discusses the details of quantum groups (quasi-triangular Hopf algebras) needed to realize the constructions in the rest of the text. Chapter XII discusses constructions derived from the bracket state model, skein theory and the Temperley-Lieb algebra. There are four appendices. The last appendix is a quick discussion of the classicality of the Crane-Yetter state sum invariant of 4-manifolds. The book ends with a list of 24 outstanding research questions.