Minimally knotted embeddings of planar graphs (Q1323433)

A graph \(\Gamma\) in \(S^ 3\) is called minimally knotted if \(\Gamma\) is knotted, but all of its proper subgraphs are unknotted. It was conjectured that any abstractly planar graph with no free edges admits a minimally knotted embedding into the 3-sphere. \textit{L. Kawauchi} [Hyperbolic imitation of 3-manifolds (Preprint 1987)] gave a proof of this conjecture, using his ``hyperbolic imitations'' method. The purpose of this paper is to give a simpler and elementary proof of this result.