3-manifolds with planar presentations and the width of satellite knots

ABSTRACT. We consider compact 3-manifolds M having a submersion h to R in which each generic point inverse is a planar surface. The stan-dard height function on a submanifold of S 3 is a motivating example. To Mh we associate a connectivity graph Γ. For M S3, 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 3M is a tree, then there is a level-preserving reimbedding of M so that S3M is a connected sum of handlebodies. Corollary: The width of a satellite knot is no less than the width of its pattern knot and so wK1#K2 maxwK1wK2 The notion of thin position, introduced by D. Gabai [G], has been em-ployed with great success in many geometric constructions. Yet the under-lying 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 K1 in thin position on top of a copy of K2 in thin position, it is easily seen that wK1#K2 wK1wK22. Here 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 there-fore 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, bK1#K2 bK1 bK2 1, see the paper [S] by H. 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 Received by the editors September 28, 2003. The authors thank RIMS Kyoto, where this work was begun, Professor Tsuyoshi Kobayashi for inviting us to RIMS, Yo’av Rieck for helpful conversations there, and NS