?-reducing Dehn surgeries and 1-bridge knots

A 3-manifold is ∂-reducible if ∂M is compressible in M . By definition, this means that there is a disk D properly embedded in M so that ∂D is an essential curve in ∂M . The disk D is called a compressing disk of ∂M , or a ∂-reducing disk of M . Now suppose M is a ∂-reducible manifold. Let K be a knot in a 3-manifold M such that ∂M is incompressible in M − K. A Dehn surgery on K is called ∂-reducing if the surgered manifold is ∂-reducible. It is known that in generic case there are very few ∂reducing surgeries. More precisely, if there is no essential annulus in M− Int N(K) with one boundary component in both ∂M and ∂N(K), then there are at most three ∂-reducing surgeries [12]. Here N(K) denotes a regular neighborhood of K in M . Examples of Berge [1, 2] and Gabai [4] show that “three” is the best possible in general. However, all the examples in the above papers are 1-bridge knots. Intuitively, if a knot is “very knotted” in a manifold, then it should admit no ∂-reducing surgeries. This leads to the 1-bridge problem of Berge, which asks if 1-bridge knots are the only ones that admit nontrivial ∂-reducing surgeries. To be more precise, we need the following definitions.