Distance between toroidal surgeries on hyperbolic knots in the 3-sphere

Abstract

For a hyperbolic knot in the 3 3 -sphere, at most finitely many Dehn surgeries yield non-hyperbolic 3 3 -manifolds. As a typical case of such an exceptional surgery, a toroidal surgery is one that yields a closed 3 3 -manifold containing an incompressible torus. The slope corresponding to a toroidal surgery, called a toroidal slope, is known to be integral or half-integral. We show that the distance between two integral toroidal slopes for a hyperbolic knot, except the figure-eight knot, is at most four.