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

TL;DR

This paper studies 3-manifolds with planar surface fibrations and explores how their connectivity graphs relate to embeddings, with applications to the width of satellite knots and their patterns.

Contribution

It introduces a framework linking connectivity graphs of 3-manifolds to their embeddings and applies this to establish bounds on the width of satellite knots.

Findings

Connectivity graphs are trees if and only if certain embeddings exist.

The width of a satellite knot is at least the width of its pattern knot.

The width of a connected sum of knots is at least the maximum of the individual widths.

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 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.