Thin position and planar surfaces for graphs in the 3-sphere

TL;DR

This paper extends Thompson's theorem to trivalent graphs in the 3-sphere, showing that their complements contain essential planar surfaces or are in bridge position, ensuring the existence of useful planar surfaces.

Contribution

It generalizes a theorem from knots to graphs, demonstrating the presence of essential planar surfaces in graph complements in the 3-sphere.

Findings

Graph complements contain essential almost meridional planar surfaces or are in bridge position.

Extension of Thompson's theorem from knots to graphs.

Any graph complement in S^3 contains a useful planar surface.

Abstract

We show that given a trivalent graph in $S^3$, either the graph complement contains an essential almost meridional planar surface or thin position for the graph is also bridge position. This can be viewed as an extension of a theorem of Thompson to graphs. It follows that any graph complement always contains a useful planar surface.