Narrow Results

In this paper, we define edge sum of some trivializable graphs. The main theorem of this paper is that an edge sum of some trivializable graphs at handle edges is trivializable and using this theorem we actually construct a larger class of trivializable graphs.

A planar graph is said to be trivializable if every regular projection of the graph produces a trivial spatial embedding by giving some over/under informations to the double points. Every minor of a trivializable graph is also trivializable, thus the set of forbidden graphs is finite. Seven forbidden graphs for the trivializability were previously known. In this paper, we exhibit nine more forbidden graphs.

It is known that the unknotting number $u(L)$ of a link $L$ is less than or equal to half the crossing number $c(L)$ of $L$. We show that there are a planar graph $G$ and its spatial embedding $f$ such that the unknotting number $u(f)$ of $f$ is greater than half the crossing number $c(f)$ of $f$. We study relations between unknotting number and crossing number of spatial embedding of a planar graph in general.