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A remark on companionship and property P. (English) Zbl 0602.57006
A theorem of C. McA. Gordon [Trans. Am. Math. Soc. 275, 687-708 (1983; Zbl 0519.57005)] says that, if some non-trivial Dehn surgery on a nontrivial knot K yields a homotopy 3-sphere, then the same is true for some nontrivial simple knot L (i.e. L has no proper companions).
Using a recent result of D. Gabai on the effect of Dehn filling on Thurston norm minimizing surfaces in 3-manifolds, this note gives a short proof of the above theorem in a stronger form, in which L can be taken to be a companion of K.
Reviewer: J.Howie

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
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[1] David Gabai, Foliations and genera of links, Topology 23 (1984), no. 4, 381 – 394. · Zbl 0567.57021 · doi:10.1016/0040-9383(84)90001-6 · doi.org
[2] C. McA. Gordon, Dehn surgery and satellite knots, Trans. Amer. Math. Soc. 275 (1983), no. 2, 687 – 708. · Zbl 0519.57005
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