Scharlemann, Martin A remark on companionship and property P. (English) Zbl 0602.57006 Proc. Am. Math. Soc. 98, 169-170 (1986). 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 Cited in 1 Document MSC: 57K10 Knot theory 57K30 General topology of 3-manifolds Keywords:property P; Dehn surgery; homotopy 3-sphere; Dehn filling on Thurston norm minimizing surfaces Citations:Zbl 0519.57005 × Cite Format Result Cite Review PDF Full Text: DOI References: [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 [2] C. McA. Gordon, Dehn surgery and satellite knots, Trans. Amer. Math. Soc. 275 (1983), no. 2, 687 – 708. · Zbl 0519.57005 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.