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A classification is given for the composite knots and the Dehn surgery on these knots which yield Seifert fibered surgery manifolds. We prove that if a knot K is the composition of two torus knots, then some (unique) integral surgery on K yields a Seifert fibered manifold, and conversely if the surgery manifold of a composite knot K is Seifert fibered, then K is the composition of two torus knots and the surgery must be integral surgery, which is uniquely determined. In this paper we classify the composite knots and the Dehn surgeries on these knots which yield Seifert fibered surgery manifolds. In [8] Moser conjectured that surgery on a nontorus knot could not yield a Seifert fibered manifold, in particular, could not yield a lens space. Many counterexamples to Moser's conjecture have been found. Baily and Rolfsen [2] showed that the lens space L(23, 7) could be obtained by -23-surgery on the (11, 2)-cable knot about the trefoil knot; Simon discovered similar examples. Fintushel and Stern [4] constructed infinitely many noniterated torus knots, upon which certain surgery yields lens spaces. More recently, Gordon [6] classified the surgery manifolds of all iterated torus knots, Berge [3] and Gabai [5] have independently constructed an infinite collection of knots in solid tori such that certain surgery on them in the solid torus yield D2 x S , and therefore yield lens spaces when the knots are considered to be in S . Since cable knots are prime knots and the surgery manifold of a composite knot contains an incompressible torus (see, for example, [6]), none of the above-mentioned knots is a composite knot. Indeed, no nontrivial surgery on a composite knot may yield a lens space. It is then natural to ask if any such surgery manifold is Seifert fibered. In this paper, we show that if a knot K is the composition of two torus knots, then some (unique) integral surgery on K yields a Seifert fibered manifold, and conversely if a surgery manifold of a composite knot K is Seifert fibered, then the surgery Received by the editors January 4, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M05, 57M25.

Seifert fibered surgery manifolds of composite knots