Non-invertible knots having toroidal Dehn surgery of hitting number four

Abstract

We show that there exist infinitely many non-invertible, hyperbolic knots that admit toroidal Dehn surgery of hitting number four. The resulting toroidal manifold contains a unique incompressible torus meeting the core of the attached solid torus in four points, but no incompressible torus meeting it less than four points. For a hyperbolic knot in the 3-sphere S 3 , at most finitely many Dehn surgeries yield non-hyperbolic 3-manifolds by Thurston's hyperbolic Dehn surgery theorem. Such Dehn surgeries are called exceptional Dehn surgeries. A typical one is Dehn surgery creating an incompressible torus, called toroidal Dehn surgery. By Gordon-Luecke (7, 8), the surgery slope of toroidal Dehn surgery is integral or half-integral, and the latter happens only for Eudave- Munoz knots (3). Thus the study of integral toroidal Dehn surgery is the next challenging task. We now introduce the notion of hitting number for toroidal Dehn surgery. Let K be a hyperbolic knot in S 3 . Suppose that the resulting 3-manifold by r-Dehn surgery, denoted by KðrÞ, is toroidal. For an incompressible torus T contained in KðrÞ, let jKV Tj denote the number of points in KV T, where Kis the core of the attached solid torus of KðrÞ. For a pair ðK; rÞ, we call minfjKV Tj : T is an incompressible torus in KðrÞg the hitting number of ðK; rÞ. Since K is hyperbolic, a hitting number is positive. This is a natural measure of the complexity of toroidal Dehn surgery. We should mention that the only possible odd hitting number is one.