All integral slopes can be Seifert fibered slopes for hyperbolic knots

Abstract

Which slopes can or cannot appear as Seifert fibered slopes for hyperbolic knots in the $3$–sphere $S^3$? It is conjectured that if $r$–surgery on a hyperbolic knot in $S^3$ yields a Seifert fiber space, then $r$ is an integer. We show that for each integer $n\in \mathbb{Z}$, there exists a tunnel number one, hyperbolic knot $K_n$ in $S^3$ such that $n$–surgery on $K_n$ produces a small Seifert fiber space.