The realization problem for non-integer Seifert fibered surgeries

Abstract

Conjecturally, the only knots in S 3 with non-integer surgeries producing Seifert fibered spaces are torus knots and cables of torus knots. In this paper, we make progress on the associated realization problem. Let Y be a small Seifert fibered space bounding a positive definite plumbing with central vertex of weight e such that Y arises by non-integer p/q-surgery on a knot in S 3 . We show that if e ≥ 2 and the slope p/q is negative, or e ≥ 3 and p/q is positive, then Y can be obtained by p/q-surgery on a torus knot or a cable of a torus knot.