By Vladimir G Turaev
Because of the powerful charm and vast use of this monograph, it truly is now on hand in its moment revised variation. The monograph supplies a scientific remedy of third-dimensional topological quantum box theories (TQFTs) in accordance with the paintings of the writer with N. Reshetikhin and O. Viro. This topic used to be encouraged via the invention of the Jones polynomial of knots and the Witten-Chern-Simons box idea. at the algebraic facet, the learn of third-dimensional TQFTs has been inspired by way of the speculation of braided different types and the idea of quantum teams. The ebook is split into 3 elements. half I offers a development of three-dimensional TQFTs and 2-dimensional modular functors from so-called modular different types. this provides an enormous classification of knot invariants and 3-manifold invariants in addition to a category of linear representations of the mapping type teams of surfaces. partly II the means of 6j-symbols is used to outline country sum invariants of 3-manifolds. Their relation to the TQFTs built partly I is proven through the speculation of shadows. half III presents structures of modular different types, according to quantum teams and skein modules of tangles within the 3-space. This basic contribution to topological quantum box thought is on the market to graduate scholars in arithmetic and physics with wisdom of simple algebra and topology. it truly is an imperative resource for everybody who needs to go into the leading edge of this interesting zone 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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Additional resources for Quantum Invariants of Knots and 3-Manifolds
11 2. Let G be a multiplicative abelian group, K a commutative ring with unit, c a bilinear pairing G G ! K where K is the multiplicative group of invertible elements of K. Thus c(gg0 ; h) = c(g; h) c(g0 ; h) and c(g; hh 0 ) = c(g; h) c(g; h 0 ) for any g; g0 ; h; h 0 2 G. Using these data, we construct a ribbon category V. The objects of V are elements of G. For any g 2 G, the set of morphisms g ! g is a copy of K. For distinct g; h 2 G the set of morphisms g ! h consists of one element called zero.
To see this, we ˇrst deform the graph so that its coupons lie in a standard position and then we deform the bands so that they go \parallel" to the plane of the picture. The only problem which we 34 I. 2 may encounter is that the bands may be twisted several times around their cores. However, both positive and negative twists in a band are isotopic to curls which go \parallel" to the plane. 3 which presents positive and negative twists in a band. ) Note that positivity of the twist does not depend on the direction of the band and depends solely on the orientation of the ambient 3-manifold; we use everywhere the right-handed orientation in R3 .
For any projective K-module V, set V = V ? = HomK (V; K) and deˇne d V to be the evaluation pairing v ˝ w 7! v(w) : V ? ˝ V ! K. Finally, deˇne bV to be the homomorphism K ! V ˝ V ? dual to d V : V ? ˝ V ! K where we use the standard identiˇcations K ? = K and (V ? ˝ V)? = V ?? ˝ V ? = V ˝ V ? The last two equalities follow from projectivity of V. ) All axioms of ribbon categories are easily seen to be satisˇed. c) is an exercise in linear algebra, it is left to the reader. The ribbon category Proj(K) is not interesting from the viewpoint of applications to knots.
Quantum Invariants of Knots and 3-Manifolds by Vladimir G Turaev