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By Tomotada Ohtsuki

An in depth and self-contained presentation of quantum and comparable invariants of knots and 3-manifolds. Polynomial invariants of knots, similar to the Jones and Alexander polynomials, are built as quantum invariants, in different phrases, invariants derived from representations of quantum teams and from the monodromy of suggestions to the Knizhnik-Zamolodchikov equation. With the advent of the Kontsevich invariant and the speculation of Vassiliev invariants, the quantum invariants develop into well-organized. Quantum and perturbative invariants, the LMO invariant, and finite kind invariants of 3-manifolds are mentioned. The Chern-Simons box thought and the Wess-Zumino-Witten version are defined because the actual heritage of the invariants.

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Additional info for Quantum Invariants: A Study of Knot, 3-Manifolds, and Their Sets

Example text

In particular, the operator invariant [L] of a link L belongs End(C) = C. Proof. 4. 4) are obtained by the construction of the bracket [D\. 19). 5) is obtained in the same way. 21), and the equality R • R^1 = idy (8) id v. • By construction, an operator invariant preserves the structure of tensor product and composition as [Ti®T2] = [Ti]®[r 2 ], [Ti • T2] = pi] • [T2]. 4 we obtain an operator invariant of framed tangles by the following theorem. 5. Let T be a framed tangle and D a sliced diagram presenting T by blackboard framing.

4) the condition is presented by (n idy)(idv u) = idy = (idy n)(u idy). (3-17) This relation is computed as follows. 17) is computed e/c idy®u ^ " ^ l3 ^nkiU^ej. u ek This is equal to ek if the matrices (u l J ) and (n^j) are the inverses of each other. 17) we have that u®idK \ ~ ^ ii ^ ,0, idvigm v - ^ ii e/t i—> 2_^u Jei ® ej ® e i—> ^ k M uJn^i- *>j This is equal to efc if the above condition holds. 17) if the matrices (n^) and (ulJ) are the inverses of each other. 17) such that the matrix (u1^) is the inverse of the matrix (rijj).

Further, we introduce the Turaev moves among them, which play a role similar to the Reidemeister moves among link diagrams and the Markov moves among braids. By using sliced diagrams of tangles we give rigorous constructions of operator invariants of tangles. In particular, we obtain reconstructions of the Jones and Alexander polynomials as operator invariants. 1 Tangles and their sliced diagrams In this section we introduce sliced diagrams by expressing tangles by elementary tangle diagrams. To describe isotopy classes of tangles (in particular, links) we introduce the Turaev moves among sliced diagrams.

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