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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Symmetry of knots and cyclic surgery

Authors: Shi Cheng Wang and Qing Zhou
Journal: Trans. Amer. Math. Soc. 330 (1992), 665-676
MSC: Primary 57M25; Secondary 57N12
MathSciNet review: 1031244
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Abstract: If a nontorus knot $ K$ admits a symmetry which is not a strong inversion, then there exists no nontrivial cyclic surgery on $ K$. No surgery on a symmetric knot can produce a fake lens space or a $ 3$-manifold $ M$ with $ \vert{\pi _1}(M)\vert= 2$. This generalizes the result of Culler-Gordon-Luecke-Shalen-Bleiler-Scharlemann and supports the conjecture that no nontrivial surgery on a nontrivial knot yields a $ 3$-manifold $ M$ with $ \vert{\pi _1}(M)\vert < 5$.

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