C-essential surfaces in (3-manifold, graph) pairs

Abstract

Let $T$ be a graph in a compact, orientable 3--manifold $M$ and let $\Gamma$ be a subgraph. $T$ can be placed in bridge position with respect to a Heegaard surface $H$. We show that if $H$ is what we call $(T,\Gamma )$-c-weakly reducible in the complement of $T$ then either a "degenerate" situation occurs or $H$ can be untelescoped and consolidated into a collection of "thick surfaces" and "thin surfaces". The thin surfaces are c-essential (c-incompressible and essential) in the graph exterior and each thick surface is a strongly irreducible bridge surface in the complement of the thin surfaces. This strengthens and extends previous results of Hayashi-Shimokawa and Tomova to graphs in 3-manifolds that may have non-empty boundary.

33 pages, 9 figures. Accepted for publication in Communications in Analysis and Geometry

Cited by in corpus (1)

• Thin position for knots, links, and graphs in 3-manifolds