The Uniqueness of Skeleton Presentation of Complete Bipartite Graph $K_{m,n}$

Abstract:

      Kobayashi discussed some kinds of standard embeddings into 3-manifolds of spatial graphs. He introduced the concept of book presentation, which is a standard embedding of spatial graphs with good properties, and proved that the book presentation of minimum sheets of $K_n$ is unique up to the sheet translation and the ambient isotopy. In this present paper we give the definition of skeleton presentation of spatial graphs, and prove that the skeleton presentation of minimum floors of a complete bipartite graph $K_{m,n}$ is unique up to ambient isotopy.