EVERY LINK IS A BOUNDARY OF A COMPLETE BIPARTITE GRAPH $K_{2,n}$

Abstract

A voltage assignment on a graph was used to enumerate all possible 2-cell embeddings of a graph onto surfaces. The boundary of the surface which is obtained from 0 voltage on every edges of a very special diagram of a complete bipartite graph $K_{m,n}$ is surprisingly the $(m, n)$ torus link. In the present article, we prove
that every link is the boundary of a complete bipartite multi-graph
$K_{m,n}$ for which voltage assignments are either −1 or 1 and that every link is the boundary of a complete bipartite graph $K_{2,n} for which voltage assignments are either −1, 0 or 1 where edges in the diagram of graphs may be linked but not knotted.