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 way of the invention of the Jones polynomial of knots, the Witten-Chern-Simons box conception, and the speculation of quantum teams. the writer, one of many major specialists within the topic, supplies a rigorous and self-contained exposition of basic algebraic and topological innovations that emerged during this conception
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Extra info for Quantum Invariants of Knots and 3-Manifolds
The bands and annuli of the graph should go close and \parallel" to this plane. The projections of the cores of bands and annuli in the plane R 0 R should have only double transversal crossings and should not overlap with the projections of coupons. After having deformed the graph in such a position we draw the projections of the coupons and the cores of the bands and annuli in R 0 R taking into account the overcrossings and undercrossings of the cores. The projections of the cores of bands and annuli are oriented in accordance with their directions.
Ribbon graphs over V. Fix a strict monoidal category with duality V. A ribbon graph is said to be colored (over V) if each band and each annulus of the graph is equipped with an object of V. This object is called the color of the band (the annulus). The coupons of a ribbon graph may be colored by morphisms in V. Let Q be a coupon of a colored ribbon graph ˝. 4 where Q is oriented counterclockwise). Let W 1 ; : : : ; W n be the colors of the bands of ˝ incident to the top base of Q and encountered in the order induced by the opposite orientation of Q.
Indeed, we have c V;W = (c W;V ) 1 so that the morphisms associ- 28 I. 10 ated with diagrams are preserved under trading overcrossings for undercrossings, which kills the 3-dimensional topology of diagrams (cf. 6). 5 to the morphisms and objects of Proj(K) we get the notions of a dimension for projective K-modules and a trace for Kendomorphisms of projective K-modules. We shall denote these dimension and trace by Dim and Tr respectively. They generalize the usual dimension and trace for free modules and their endomorphisms (cf.
Quantum Invariants of Knots and 3-Manifolds by Vladimir G Turaev