Toroidal surgeries on hyperbolic knots

Abstract

For a hyperbolic knot $K$ in $S^3$, a toroidal surgery is Dehn surgery which yields a $3$-manifold containing an incompressible torus. It is known that a toroidal surgery on $K$ is an integer or a half-integer. In this paper, we prove that all integers occur among the toroidal slopes of hyperbolic knots with bridge index at most three and tunnel number one.