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

Abstract

Given a knot K in a 3-manifold M, we use N(K) to denote a regular neighbourhood of K. Suppose γ is a slope (i.e. an isotopy class of essential simple closed curves) on ∂N(K). The surgered manifold along γ is denoted by (H, K; γ), which by definition is the manifold obtained by gluing a solid torus to H – Int N(K) so that γ bounds a meridional disc. We say that M is ∂-reducible if ∂M is compressible in M, and we call γ a ∂-reducing slope of K if (H, K; γ) is ∂-reducible. Since incompressible surfaces play an important rôle in 3-manifold theory, it is important to know what slopes of a given knot are ∂-reducing. In the generic case there are at most three ∂-reducing slopes for a given knot [12], but there is no known algorithm to find these slopes. An exceptional case is when M is a solid torus, which has been well studied by Berge, Gabai and Scharlemann [1, 4, 5, 10]. It is now known that a knot in a solid torus has ∂-reducing slopes only if it is a 1-bridge braid. Moreover, all such knots and their corresponding ∂-reducing slopes are classified in [1]. For 1-bridge braids with small bridge width, a geometric method of detecting ∂-reducing slopes has also been given in [5]. It was conjectured that a similar result holds for handlebodies, i.e. if K is a knot in a handlebody with H – K ∂-irreducible, then K has ∂-reducing slopes only if K is a 1-bridge knot (see below for definitions). One is referred to [13] for some discussion of this conjecture and related problems.