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Closed 3-manifolds unchanged by Dehn surgery. (English) Zbl 0771.57008

Let \(M\) be a compact orientable 3-manifold with \(\partial M\) an incompressible torus, and let \(M(s)\) be the closed manifold obtained by Dehn filling along the simple closed curve (“slope”) \(s\subset \partial M\). The generalized knot complement problem asks whether if \(M(r)\) and \(M(s)\) are orientation preserving homeomorphic then \(r\) and \(s\) must be isotopic. This is so if \(M(s)\cong S^ 3\), by the work of C. McA. Gordon and J. Luecke [J. Am. Math. Soc. 2, No. 2, 371-415 (1989; Zbl 0678.57005)]. Here it is shown that if \(M\) is the complement of a 2- bridge torus knot there are infinitely many pairs of distinct slopes such that the corresponding Dehn fillings are Seifert fibred with 3 exceptional fibres and are homeomorphic via orientation reversing maps. If the knot is the trefoil knot we may also assume that the Dehn fillings have finite fundamental group.

MSC:

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

Citations:

Zbl 0678.57005
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