Only single twists on unknots can produce composite knots

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.