Narrow Results

Given a simple manifold, we investigate two Dehn fillings, one of which contains a non-separating sphere and the other contains an essential torus.

For a hyperbolic knot in the 3-sphere, the distance between toroidal surgeries is at most 5, except the figure eight knot. In this paper, we determine all hyperbolic knots that admit two toroidal surgeries with distance 5.

We show an infinite family of satellite knots that can be unknotted by a single band move, but such that there is no band unknotting the knots which is disjoint from the satellite torus.