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The classification of exceptional Dehn surgeries on 2-bridge knots. (English) Zbl 0964.57013

In the unpublished but well-circulated W. Thurston notes on 3-manifolds, all of the manifolds resulting from Dehn surgery on a figure eight knot are analyzed in order to illustrate that the most 3-manifolds admit a hyperbolic metric. In fact, all but 10 surgeries result in a hyperbolic 3-manifold. Each of the exceptional surgeries result in either \(S^3\), a reducible manifold, a toroidal manifold, or a Seifert fiber space. In this paper, the authors extend the same analysis from the figure eight knot to the class of all 2-bridge knots. In order to do this they bring together many different results about 3-manifolds. Their paper relies heavily on the paper [A. E. Hatcher and W. Thurston, Invent. Math. 79, 225-246 (1985; Zbl 0602.57002)]. It uses C. Delman’s construction of essential braided surfaces [Topology Appl. 63, No. 3, 201-221 (1995; Zbl 0873.57007)], R. Roberts’s construction of essential branched surfaces [Comment. Math. Helv. 70, No. 4, 516-545 (1995; Zbl 0855.57009)] and work of each author individually [M. Brittenham, Topology 37, No. 3, 665-672 (1998; Zbl 0912.57010); Y.-Q. Wu, Trans. Am. Math. Soc. 351, No. 6, 2275-2294 (1999; Zbl 0919.57008)].

MSC:

57M50 General geometric structures on low-dimensional manifolds
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
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