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Surgery on a knot in (surface \(\times I\)). (English) Zbl 1197.57011

In the paper under review the authors prove a generalisation of two results which follow from work of D. Gabai [Topology 28, No.1, 1-6 (1989; Zbl 0678.57004)]. Roughly speaking, Gabai proved that if \(A\) is an annulus and \(K\) is a knot in \(A\times I\) such that after a nontrivial Dehn surgery on \(K\) the annulus \(A\times \{ 0\} \) compresses, then \(K\) is parallel in \(A\times I\) to the core curve \(\alpha\) of \(A\times \{0\}\) and the annulus that describes the parallelism determines the slope of the Dehn surgery. He also proved a similar result considering a torus. The generalisation says the following: Suppose \(F\) is a compact orientable surface, \(K\) a knot in \(F \times I\), and \((F\times I)_{surg} \) is the 3-manifold obtained by some nontrivial surgery on \(K\). If \(F\times \{0\}\) compresses in \((F\times I)_{\text{surg}}\) then \(K\) is parallel to an essential simple closed curve in \(F\times \{0\}\). Moreover, the annulus that describes the parallelism determines the slope of the surgery. Also, as an application the authors prove that if \((F\times I)_{\text{surg}}\) is reducible then either \(K\) lies in a 3-ball, or \(K\) is cabled and the surgery slope is that of the cabling annulus, or \(F\) is a torus, \(K\) is parallel to an essential simple closed curve in \(F\times \{0\}\), and the annulus that describes the parallelism determines the surgery slope.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)

Citations:

Zbl 0678.57004
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Full Text: DOI arXiv

References:

[1] S Boyer, C M Gordon, X Zhang, Dehn fillings of large hyperbolic \(3\)-manifolds, J. Differential Geom. 58 (2001) 263 · Zbl 1042.57007
[2] M Culler, C M Gordon, J Luecke, P B Shalen, Dehn surgery on knots, Ann. of Math. \((2)\) 125 (1987) 237 · Zbl 0633.57006
[3] D Gabai, Foliations and the topology of \(3\)-manifolds, J. Differential Geom. 18 (1983) 445 · Zbl 0539.57013
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[8] M Scharlemann, Producing reducible \(3\)-manifolds by surgery on a knot, Topology 29 (1990) 481 · Zbl 0727.57015
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