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We provide a new proof of the following results of H. Schubert: If K is a satellite knot with companion J and pattern (V , L) with index k, then the bridge numbers satisfy the following: b(K) ≥ k · (b(J)). In addition, if K is a composite knot with summands J and L, then bIn "Über eine numerische Knoteninvariante" [1], Horst Schubert proved that for a satellite knot K with companion J and pattern of index k, bridge numbers satisfy the inequality b(K) ≥ k · (b(J)). He also proved that for a composite knot K with summands J and L, the bridge numbers satisfy bHis investigation was motivated by the question as to whether a knot can have only finitely many companions. Together with the fact that the only bridge number one knot is the unknot, his result showed that the answer to this question is yes.Schubert's main result may be recovered by a much shorter proof. This shorter proof grew out of an endeavour to recast the problem within the framework of the thin position of a knot. This framework turns out to be far more refined than necessary. The proof here does not employ the notion of thin position. It does, however, rely heavily on the idea of rearranging the order in which critical points occur to suit one's purpose, an idea fundamental to the notion of thin position of knots and 3-manifolds. In this way it differs dramatically from Schubert's proof. It also differs from Schubert's in that it relies on the consideration of Morse functions on S 3 whose level sets are spheres (except for the maximum and minimum) and their induced foliations. This streamlines the terminology and the complexity of the argument. Schubert's proofs of the results reproven here involve 25 pages containing 15 lemmas which involve a consideration of up to three cases.

Abstract.We give the classification theorem for Heegaard splittings of fiberwise orientable Seifert fibered spaces with nonempty boundary. A thin position argument yields a reducibility result which, by induction, shows that all Heegaard splittings of such manifolds are vertical in the sense of Lustig-Moriah. Algebraic arguments allow a classification of the vertical Heegaard splittings.

Abstract. 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 S 3 is a motivating example. To (M, h) we associate a connectivity graph Γ. For M ⊂ S 3 , Γ 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 S 3 − M is a tree, then there is a level-preserving reimbedding of M so that S 3 − M is a connected sum of handlebodies. Corollary.• The width of a satellite knot is no less than the width of its pattern knot and soThe notion of thin position, introduced by D. Gabai [G], has been employed with great success in many geometric constructions. Yet the underlying notion of the width of a knot remains shrouded in mystery. Little is known about the width of specific knots, or how knot width behaves under connected sum. By stacking a copy of K 1 in thin position on top of a copy of K 2 in thin position, it is easily seen thatHere we establish a lower bound for the width of a knot sum: the width is bounded below by the maximum of the widths of its summands and therefore also by one-half the sum of the widths of its summands.Knot width can be thought of as a kind of refinement of bridge number. Interest in how the width of a knot behaves under connected sum is inspired, in part, by the fact that bridge number behaves very well. Indeed for bridge number, Schubert, or [Sch] for a much shorter proof. The shorter proof in [Sch] crystallized out of an investigation into whether or not thin position arguments clarify the behaviour of bridge number under connected sum. The answer to that question appears to be no: width seems to be a much more refined invariant than can be useful for the recovery of Schubert's result. In particular, the argument in [Sch] fails in settings where the swallow follow torus is too convoluted. One suspects that degeneration of width under connected sum of knots is possible, i.e., that there might be knots K 1 , K 2 , such that w(K 1 #K 2 ) < w(K 1 ) + w(K 2 ) − 2. The situation may be analogous to that of