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We consider compact 3-manifolds M having a submersion h to R in which each generic point inverse is a planar surface. The standard height function on a submanifold of the 3-sphere is a motivating example. To (M, h) we associate a connectivity graph G. For M in the 3-sphere, G is a tree if and only if there is a Fox reimbedding of M which carries horizontal circles to a complete collection of complementary meridian circles. On the other hand, if the connectivity graph of the complement of M is a tree, then there is a level-preserving reimbedding of M so that its complement is a connected sum of handlebodies. Corollary: The width of a satellite knot is no less than the width of its pattern knot. In particular, the width of K_1 # K_2 is no less than the maximum of the widths of K_1 and K_2., Comment: 23 pages, 13 figures

Let K be a knot in a closed orientable irreducible 3-manifold M. Suppose M admits a genus 1 Heegaard splitting and we denote by H the splitting torus. We say H is a 1-genus 1-bridge splitting of (M, K) if H intersects K transversely in two points, and divides (M, K) into two pairs of a solid torus and a boundary parallel arc in it. It is known that a 1-genus 1-bridge splitting of a satellite knot admits a satellite diagram disjoint from an essential loop on the splitting torus. If M = S 3 and the slope of the loop is longitudinal in one of the solid tori, then K is obtained by twisting a component of a 2-bridge link along the other component. We give a criterion for determining whether a given 1-genus 1-bridge splitting of a knot admits a satellite diagram of a given slope or not. As an application, we show there exist counter examples for a conjecture of Ait Nouh and Yasuhara.

We get a necessary condition under which a nonsimple knot (i.e. a satellite knot) admits a nontrivial surgery producing a lens space. Interesting corollaries are: ( 1) if a lens space can be obtained from a nontrivial surgery on a nonsimple knot, then the order of its fundamental group is not smaller than 23; (2) any nonsimple knot admits at most one nontrivial surgery which produces a lens space. Some topologists are interested in the problem of doing surgery on a knot complement to get a lens space. This note is devoted to providing more information on this problem. Our main result is the following: Theorem 1. If a lens space can be obtainedfrom a nontrivial surgery on a nonsimple knot (i.e. a satellite knot) J(K), then K is a torus knot and J is a closed braid in a tubular neighborhood of K. There are two interesting corollaries of Theorem 1. D. Gabai proved that one cannot get a lens space with fundamental group of order 1 (i.e. S3) from a nontrivial surgery on a nonsimple knot (in fact nonsimple knots have property P). J. Bailey and D. Rolfsen [BR] (also R. Fintushel and R. Stern [FS] and C. Gordon [G]) proved that some lens space with fundamental group of order 23 can be obtained from a nontrivial surgery on a nonsimple knot. The first corollary says that 23 is the smallest possible. Corollary 1. If a lens space can be obtained from a nontrivial surgery on a nonsimple knot (i.e. a satellite knot), then the order of its fundamental group is not smaller than 23. M. Culler, C. M. Gordon, J. Luecke, and P. Shalen [CGLS] proved that there are at most two nontrivial surgeries on a nontorus knot which produce lens spaces. Corollary 2 below says that a nontorus knot which admits two nontrivial surgeries producing lens spaces must be a hyperbolic knot. (Berge found infinitely many such nontorus knots.) Received by the editors February 4, 1988, and in revised form, August 17. 1988. 1980 Alathernatics S/ihject Classification (I1985 Rewision). Primary 57N 10. (? 1989 American Mathematical Society 0002-9939/89 $ 1.00 + $.25 per page

For certain kinds of 3 3 -manifolds, the question whether such a manifold can be obtained by nontrivial Dehn surgery on a knot in S 3 {S^3} is reduced to the corresponding question for hyperbolic knots. Examples are, whether one can obtain S 3 {S^3} , a fake S 3 {S^3} , a fake S 3 {S^3} with nonzero Rohlin invariant, S 1 × S 2 {S^1} \times {S^2} , a fake S 1 × S 2 , S 1 × S 2 # M {S^1} \times {S^2}, {S^1} \times {S^2} \# M with M M nonsimply-connected, or a fake lens space.

This answers a question apparently first raised by Tietze [T, p. 83]. It was previously known that there were at most two knots with a given complement [CGLS, Corollary 3]. The notion of equivalence of knots can be strengthened by saying that K and K' are isotopic if the above homeomorphism h is isotopic to the identity, or, equivalently, orientation-preserving. The analog of Theorem 1 holds in this setting too: if two knots have complements that are homeomorphic by an orientation-preserving homeomorphism, then they are isotopic. Theorem 1 and its orientation-preserving version are easy consequences of the following theorem concerning Dehn surgery.

We show that a knot exists with the property that there exists a killer of the knot group which is not the image of the meridian under any automorphism.

In [D], Dehn considered the following method for constructing 3-manifolds: remove a solid torus neighborhood N(K) of some knot X in the 3-sphere S and sew it back differently. In particular, he showed that, taking X to be the trefoil, one could obtain infinitely many non-simply-connected homology spheres o in this way. Let Mx = S —N(K). Then the different resewings are parametrized by the isotopy class r of the simple closed curve on the torus oMK that bounds a meridional disk in the re-attached solid torus. We denote the resulting closed oriented 3-manifold by MK(), and say that it is obtained by r-Dehn surgery on X. More generally, one can consider the manifolds ML(*) obtained by r-Dehn surgery on a fc-component link L = Xi U • • • U Kk in S, where r = (r\,..., ;>). It turns out that every closed oriented 3-manifold can be constructed in this way [Wal, Lie]. Thus a good understanding of Dehn surgery might lead to progress on general questions about the structure of 3-manifolds. Starting with the case of knots, it is natural to extend the context a little and consider the manifolds M(r) obtained by attaching a solid torus V to an arbitrary compact, oriented, irreducible (every 2-sphere bounds a 3-ball) 3-manifold M with dM an incompressible torus, where r is the isotopy class (slope) on dM of the boundary of a meridional disk of V. We say that M(r) is the result of r-Dehn filling on M. An observed feature of this construction is that