Roots of knotted graphs and orbifolds

TL;DR

This paper introduces a method to decompose graphs in 3-manifolds into unique roots via admissible sphere compressions, providing a new proof of prime decompositions of 3-orbifolds.

Contribution

It establishes the existence and uniqueness of roots for pairs (M,G) and applies this to simplify proofs of orbifold prime decompositions.

Findings

Existence of roots for any pair (M,G)

Uniqueness of the root decomposition

Simplified proof of Petronio's theorem

Abstract

Let G be a graph in a 3-manifold M. We compress the pair (M,G) along admissible 2-spheres as long as possible. What we get is a root of (M,G). Our main result is that for any pair (M,G) the root exists and is unique. As a corollary we get an easy proof of Petronio's theorem on prime decompositions of 3-orbifolds.