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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)
##### Keywords:
Dehn surgery; taut sutured manifold
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##### References:
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