$3$-manifolds with planar presentations 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 $S^{3}$ is a motivating example. To $(M, h)$ we associate a connectivity graph $\Gamma$. For $M \subset S^{3}$, $\Gamma$ 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.

$\bullet$ The width of a satellite knot is no less than the width of its pattern knot and so

$\bullet$ $w(K_{1} \char93 K_{2}) \geq max(w(K_{1}), w(K_{2}))$.