Incompressible surfaces and Dehn Surgery on 1-bridge Knots in handlebodies

Given a knot K in a 3-manifold M , we use N(K) to denote a regular neighborhood 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 − IntN(K) so that γ bounds a meridianal disk. 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 rule in 3-manifold theory, it is interesting to know what slopes of a given knot are ∂-reducing. In 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 its 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. The main result of the present paper is to give an algorithm which will determine all ∂-reducing slopes for a given 1-bridge knot in a handlebody. Given a 1-bridge presentation of a knot K in a handlebody H, the Main Algorithm in Section 7 will do the following. (1). Determine if K is disjoint from some compressing disk of ∂H. If it is, then ∂H is compressible after all surgeries, so all slopes are ∂-reducing. (2). If K intersects all compressing disk of ∂H, determine if K is isotopic to a simple closed curve on ∂H. If it is, then ∂H is compressible in (H,K; γ) if and only if ∆(γ, γ 0) ≤ 1, where γ0 is the the boundary of an annulus in E(K) whose other boundary component is on ∂H. Mathematics Subject Classification: 57N10, 57M25, 57M50.