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

TL;DR

This paper extends the theory of c-essential surfaces in 3-manifolds with embedded graphs, showing how certain weakly reducible bridge surfaces can be decomposed into essential thin surfaces and strongly irreducible thick surfaces, generalizing prior results.

Contribution

It introduces a new framework for analyzing (3-manifold, graph) pairs by decomposing bridge surfaces into c-essential thin surfaces and strongly irreducible thick surfaces, extending previous work to manifolds with boundary.

Findings

If a (T,Γ)-c-weakly reducible surface exists, then a decomposition into c-essential thin surfaces and strongly irreducible thick surfaces is possible.

The decomposition generalizes previous results to manifolds with boundary.

The approach provides a structured way to analyze the topology of (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.