By Vladimir G Turaev

This monograph, now in its moment revised version, offers a scientific remedy of topological quantum box theories in 3 dimensions, encouraged by means of the invention of the Jones polynomial of knots, the Witten-Chern-Simons box thought, and the speculation of quantum teams. the writer, one of many major specialists within the topic, supplies a rigorous and self-contained exposition of primary algebraic and topological recommendations that emerged during this concept

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Quantum Invariants of Knots and 3-Manifolds

This monograph, now in its moment revised variation, offers a scientific therapy of topological quantum box theories in 3 dimensions, encouraged via the invention of the Jones polynomial of knots, the Witten-Chern-Simons box thought, and the speculation of quantum teams. the writer, one of many top specialists within the topic, provides a rigorous and self-contained exposition of primary algebraic and topological ideas that emerged during this idea

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Outline. We introduce ribbon categories forming the algebraic base of the theory presented in this book. , categories with tensor product) endowed with braiding, twist, and duality. All these notions are discussed here in detail; they will be used throughout the book. We also introduce an elementary graphical calculus allowing us to use drawings in order to present morphisms in ribbon categories. As we shall see in Section 2, each ribbon category gives rise to a kind of \topological ˇeld theory" for links in Euclidean 3-space.

These ˇxed isomorphisms should satisfy two compatibility conditions called the pentagon and triangle identities, see [Ma2]. h) in the obvious way. For instance, the category of modules over a commutative ring with the standard tensor product of modules is monoidal. The ground ring regarded as a module over itself plays the role of the unit object. Note that this monoidal category is not strict. Indeed, if U; V; and W are modules over a commutative ring then the modules (U ˝ V) ˝ W and U ˝ (V ˝ W) are canonically isomorphic but not identical.

7 projective. For any projective K-module V, the dual K-module V ? = HomK (V; K) is also projective and the canonical homomorphism V ! V ?? is an isomorphism. 1. 8 Let Proj(K) be the category of projective K-modules and K-linear homomorphisms. Provide Proj(K) with the usual tensor product over K. Set & = K. It is obvious that Proj(K) is a monoidal category. We provide this category with braiding, twist, and duality. 2. The twist is given by the identity endomorphisms of objects. For any projective K-module V, set V = V ?

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