The realization problem for non-integer Seifert fibered surgeries

Ahmad Issa, Duncan McCoy

Published 2018-10-03Version 1

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 $e\geq 2$ and the slope $p/q$ is negative, or $e\geq 3$ 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.