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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Only single twists on unknots can produce composite knots

Authors: Chuichiro Hayashi and Kimihiko Motegi
Journal: Trans. Amer. Math. Soc. 349 (1997), 4465-4479
MSC (1991): Primary 57M25
MathSciNet review: 1355073
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Abstract: Let $K$ be a knot in the $3$-sphere $S^{3}$, and $D$ a disc in $S^{3}$ meeting $K$ transversely more than once in the interior. For non-triviality we assume that $\vert K \cap D \vert \ge 2$ over all isotopy of $K$. Let $K_{n}$($\subset S^{3}$) be a knot obtained from $K$ by cutting and $n$-twisting along the disc $D$ (or equivalently, performing $1/n$-Dehn surgery on $\partial D$). Then we prove the following: (1) If $K$ is a trivial knot and $K_{n}$ is a composite knot, then $\vert n \vert \le 1$; (2) if $K$ is a composite knot without locally knotted arc in $S^{3} - \partial D$ and $K_{n}$ is also a composite knot, then $\vert n \vert \le 2$. We exhibit some examples which demonstrate that both results are sharp. Independently Chaim Goodman-Strauss has obtained similar results in a quite different method.

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Additional Information

Chuichiro Hayashi
Affiliation: Department of Mathematical Sciences, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113, Japan

Kimihiko Motegi
Affiliation: Department of Mathematics, College of Humanities & Sciences, Nihon University Sakurajosui 3-25-40, Setagaya-ku, Tokyo 156, Japan

PII: S 0002-9947(97)01628-0
Keywords: Knot, twisting, primeness, Scharlemann cycle
Received by editor(s): October 13, 1994
Received by editor(s) in revised form: August 30, 1995
Additional Notes: The first author was supported in part by Fellowships of the Japan Society for the Promotion of Science for Japanese Junior Scientists.
The second author was supported in part by Grant-in-Aid for Encouragement of Young Scientists 06740083, The Ministry of Education, Science and Culture and Nihon University Research Grant B94-0025.
Article copyright: © Copyright 1997 American Mathematical Society