Embeddings of 4-Regular Graphs into 3-Space

Embeddings of 4-regular graphs into 3-space are examined by studying graph diagrams, i.e. projections of embedded graphs to an appropriate plane. New diagrams can be constructed from the old ones by replacing graph vertices with rational tangles, and these diagrams lead to topological invariants of embedded graphs. The new invariants are calculated for some examples, in particular for classes of alternating diagrams of the figure-eight graph. As an application, it is shown that these diagrams have minimal crossing number, which gives generalizations to some of the so-called Tait conjectures.