Symmetry of knots and cyclic surgery

Wang, Shi Cheng; Zhou, Qing

doi:10.1090/S0002-9947-1992-1031244-6

Symmetry of knots and cyclic surgery
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by Shi Cheng Wang and Qing Zhou PDF

Trans. Amer. Math. Soc. 330 (1992), 665-676 Request permission

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 $|{\pi _1}(M)|= 2$. 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 $|{\pi _1}(M)| < 5$.

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dimensional geometric cone structures