# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  S Boyer, C M Gordon, X Zhang, Dehn fillings of large hyperbolic $$3$$-manifolds, J. Differential Geom. 58 (2001) 263 · Zbl 1042.57007 · euclid:jdg/1090348327  M Culler, C M Gordon, J Luecke, P B Shalen, Dehn surgery on knots, Ann. of Math. $$(2)$$ 125 (1987) 237 · Zbl 0633.57006 · doi:10.2307/1971311  D Gabai, Foliations and the topology of $$3$$-manifolds, J. Differential Geom. 18 (1983) 445 · Zbl 0539.57013 · doi:10.1090/S0273-0979-1983-15089-9  D Gabai, Foliations and the topology of $$3$$-manifolds. II, J. Differential Geom. 26 (1987) 461 · Zbl 0627.57012 · euclid:jdg/1214441487  D Gabai, Surgery on knots in solid tori, Topology 28 (1989) 1 · Zbl 0678.57004 · doi:10.1016/0040-9383(89)90028-1  C M Gordon, J Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989) 371 · Zbl 0672.57009 · doi:10.1090/S0273-0979-1989-15706-6  Y Ni, Dehn surgeries that yield fibred $$3$$-manifolds, Math. Ann. 344 (2009) 863 · Zbl 1227.57012 · doi:10.1007/s00208-008-0331-3  M Scharlemann, Producing reducible $$3$$-manifolds by surgery on a knot, Topology 29 (1990) 481 · Zbl 0727.57015 · doi:10.1016/0040-9383(90)90017-E
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.