The realization problem for non-integer Seifert fibered surgeries

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 arising by $p\mathrm{/}q$-surgery on a knot in $S^3$, where $p\mathrm{/}q$ is positive and a non-integer. Let $e$ denote the weight of the central vertex in the minimal star-shaped plumbing that $Y$ bounds. We show that if $e\le -2$ or $e\ge 3$, then $Y$ can be obtained by $p\mathrm{/}q$-surgery on a torus knot or a cable of a torus knot.