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By Vladimir G Turaev
Because of the robust allure and large use of this monograph, it truly is now to be had in its moment revised version. The monograph supplies a scientific remedy of three-dimensional topological quantum box theories (TQFTs) in accordance with the paintings of the writer with N. Reshetikhin and O. Viro. This topic was once encouraged via the invention of the Jones polynomial of knots and the Witten-Chern-Simons box thought. at the algebraic part, the learn of three-d TQFTs has been stimulated by way of the speculation of braided different types and the idea of quantum teams. The booklet is split into 3 components. half I offers a building of three-d TQFTs and 2-dimensional modular functors from so-called modular different types. this provides an enormous type of knot invariants and 3-manifold invariants in addition to a category of linear representations of the mapping classification teams of surfaces. partially II the means of 6j-symbols is used to outline country sum invariants of 3-manifolds. Their relation to the TQFTs built partially I is validated through the idea of shadows. half III offers buildings of modular different types, in line with quantum teams and skein modules of tangles within the 3-space. This basic contribution to topological quantum box concept is obtainable to graduate scholars in arithmetic and physics with wisdom of uncomplicated algebra and topology. it's an necessary resource for everybody who needs to go into the vanguard of this interesting region on the borderline of arithmetic and physics. From the contents: Invariants of graphs in Euclidean 3-space and of closed 3-manifolds Foun
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V. 18 I. Invariants of graphs in Euclidean 3-space A tensor product in a category V is a covariant functor ˝ : V V ! V which associates to each pair of objects V; W of V an object V ˝ W of V and to each pair of morphisms f : V ! V 0 , g : W ! W 0 a morphism f˝g : V ˝W ! V 0 ˝W 0 . d) idV ˝ idW = idV˝W : A strict monoidal category is a category V equipped with a tensor product and an object & = &V , called the unit object, such that the following conditions hold. f) one assumes that the right-hand sides and left-hand sides of these equalities are related by ˇxed isomorphisms.
Let ˝0 be the v-colored ribbon graph obtained from ˝ by cutting ` off along its core and coloring two newly emerging annuli with U and V. Then F(˝) = F(˝0 ). Proof. The idea of the proof is the same as in the proof of the previous corollary. Take a small vertical segment of ` directed downwards and replace it by two 2. 16 where id = idU˝V . It is obvious that this modiˇcation does not change the operator invariant. Now, pushing the upper coupon along ` until it approaches the lower coupon from below we deform our ribbon graph in the position where we may cancel these two coupons.
MacLane's coherence theorem that establishes equivalence of any monoidal category to a strict monoidal category works in the setting of ribbon categories as well (cf. 4). This enables us to focus attention on strict ribbon categories: all results obtained below for these categories directly extend to arbitrary ribbon categories. 5. Traces and dimensions. Ribbon categories admit a consistent theory of traces of morphisms and dimensions of objects. This is one of the most important features of ribbon categories sharply distinguishing them from arbitrary monoidal categories.