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Only single twists on unknots can produce composite knots

Authors:	Chuichiro Hayashi Kimihiko Motegi

Institution:	Department of Mathematical Sciences, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113, Japan Kimihiko Motegi ; Department of Mathematics, College of Humanities & Sciences, Nihon University Sakurajosui 3-25-40, Setagaya-ku, Tokyo 156, Japan

Abstract:	Let be a knot in the -sphere $S^{3}$ , and a disc in $S^{3}$ meeting transversely more than once in the interior. For non-triviality we assume that $\vert K \cap D \vert \ge 2$ over all isotopy of . Let $K_{n}$ ( $\subset S^{3}$ ) be a knot obtained from by cutting and -twisting along the disc (or equivalently, performing -Dehn surgery on $\partial D$ ). Then we prove the following: (1) If is a trivial knot and $K_{n}$ is a composite knot, then $\vert n \vert \le 1$ ; (2) if 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.

Keywords:	Knot twisting primeness Scharlemann cycle

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