Ying-Qing Wu

Incompressibility of surfaces in surgered 3-manifolds

The problem we consider in this paper was raised in [3]. Suppose T is a torus on the boundary of an orientable 3-manifold X, and S is a surface on ∂X − T which is incompressible in X. A slope γ is the isotopy class of a nontrivial simple closed curve on T . Denote by X(γ) the manifold obtained by attaching a solid torus to X so that γ is the slope of the boundary of a meridian disc. Given two slopes γ1 and γ2, we denote their (minimal) geometric intersection number by ∆(γ1, γ2).

Toroidal and Annular Dehn Fillings

Suppose

The classification of nonsimple algebraic tangles

M

Cyclic surgery and satellite knots

is a hyperbolic 3-manifold which admits two Dehn fillings

The reducibility of surgered 3-manifolds

M(r_1)

Thin position and essential planar surfaces

and

Hyperbolic manifolds and degenerating handle additions

M(r_2)

Incompressible surfaces and Dehn surgery on 1-bridge knots in handlebodies

such that

Toroidal Dehn fillings on hyperbolic 3-manifolds

M(r_1)

A generalization of the handle addition theorem

contains an essential torus and