Only Single Twists on Unknots Can Produce Composite Knots

Let K be a knot in the 3-sphere S3, and D a disc in S3 meeting K transversely more than once in the interior. For non-triviality we assume that |K ∩D| ≥ 2 over all isotopy of K. Let Kn(⊂ S3) be a knot obtained from K by cutting and n-twisting along the disc D (or equivalently, performing 1/n-Dehn surgery on ∂D). Then we prove the following: (1) IfK is a trivial knot andKn is a composite knot, then |n| ≤ 1; (2) if K is a composite knot without locally knotted arc in S3 − ∂D and Kn is also a composite knot, then |n| ≤ 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.