Homology 3-spheres which are obtained by Dehn surgeries on knots
Let K be a knot in the oriented 3-sphere \(S^ 3\) and \(H_ K\) the set of all the oriented homology 3-spheres obtained by Dehn surgeries on K. First we show that for any knot K, there exists another knot K' such that {\#}H\({}_ K\cap H_{K'}\geq 2\), where {\#}... denotes the cardinal number of..., and for arbitrary n Livingston has constructed knots \(K_ 1,...,K_ n\) such that {\#}H\({}_{K_ 1}\cap...\cap H_{K_ n}\geq 2.\) On the other hand we prove that if \(K_ 1\) and \(K_ 2\) are not ambient isotopic in \(S^ 3\) and one of them has property P then {\#}\(\{\) (m,n)\(\in {\mathbb{Z}}\times {\mathbb{Z}}:\) \(M(K_ 1;1/m)=M(K_ 2;1/n)\}\) is finite, and so {\#}H\({}_{K_ 1}\cap H_{K_ 2}\) is finite. Here we denote by M(K;1/m) the manifold obtained by Dehn surgery on K and by \(=\) orientation preservingly homeomorphic. In particular for any knots \(K_ 1\) and \(K_ 2\), if \(M(K_ 1;1/m)=M(K_ 2;1/m)\) for infinitely many integers m then \(K_ 1\) and \(K_ 2\) are ambient isotopic in \(S^ 3.\) Finally we remark that {\#}H\({}_{K_ 1}\cap H_{K_ 2}\) is finite for any distinct knots \(K_ 1\), \(K_ 2\) if and only if the ``knot type conjecture'' is true.
homology 3-spheres obtained by Dehn surgeries
scientific article; zbMATH DE number 4020006
scientific article