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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.

Reviewer: Lorena Armas-Sanabria (Mexico)

### 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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XMLCite

\textit{M. Scharlemann} and \textit{A. A. Thompson}, Algebr. Geom. Topol. 9, No. 3, 1825--1835 (2009; Zbl 1197.57011)

### References:

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[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 |

[4] | D Gabai, Foliations and the topology of \(3\)-manifolds. II, J. Differential Geom. 26 (1987) 461 · Zbl 0627.57012 |

[5] | D Gabai, Surgery on knots in solid tori, Topology 28 (1989) 1 · Zbl 0678.57004 |

[6] | C M Gordon, J Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989) 371 · Zbl 0672.57009 |

[7] | Y Ni, Dehn surgeries that yield fibred \(3\)-manifolds, Math. Ann. 344 (2009) 863 · Zbl 1227.57012 |

[8] | M Scharlemann, Producing reducible \(3\)-manifolds by surgery on a knot, Topology 29 (1990) 481 · Zbl 0727.57015 |

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