Only integral Dehn surgeries can yield reducible manifolds

Abstract

Let K be a knot in S 3 , and let K ( a / b ) denote the closed oriented 3-manifold obtained by a / b -Dehn surgery on K . (We parametrize the Dehn surgeries on K by ℚ ∩ {1/0} as in [9]; in particular, K (1/0) = S 3 .) If K is a ( p, q )-cable knot, then K ( pq ) is always reducible (see Section 3), and it is conjectured by González-Acuña and Short in [5] that these are the only examples where Dehn surgery on a knot in S 3 yields a reducible manifold. One of the results in this direction proved in [5] is that if π 1 ( K ( a/b )) is a non-trivial free product then | b | ≤ 5. We show that this may be sharpened to the assertion in the title.