Only single twists on unknots can produce composite knots

Abstract

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