Topology Seminar for 5/29/2003

Marty Scharlemann
University of California at Santa Barbara

Manifolds with planar presentation and the width of satellite knots

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 re-imbedding 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.