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).


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