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Three-manifold invariants derived from the Kauffman bracket. (English) Zbl 0771.57004

It is known that any closed oriented 3-manifold can be obtained from the 3-sphere by surgery on a framed link [W. B. R. Lickorish, Ann. Math., II. Ser. 76, 531-540 (1962; Zbl 0106.371)]. The invariants of links in \(S^ 3\) may be combined to produce topological invariants of closed oriented 3-manifolds and links in these manifolds. Prior to the reviewed paper W. B. R. Lickorish, through several articles, has given a construction of 3-manifold invariants using the one variable Kauffman bracket [L. Kauffman, Topology 26, 395-407 (1987; Zbl 0622.57004)] evaluated at \(4r\)th roots of unity, \(r\geq 3\). Continuing in this direction the authors show the evaluation of the bracket at \(2p\)th roots of unity, \(p\) odd, also give 3-manifold invariants, and prove that no other evaluations at other values lead to invariants. Furthermore, they show that non-trivial invariants can only exist for evaluations at primitive \(2p\)th roots of unity, that they exist and are essentially unique (consult the article for the precise sense).

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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