A characterization of knots in a spatial graph

For a finite graph G, let Gamma (G) be the set of all cycles of G. Suppose that for each gamma is an element of Gamma (G), an embedding phi (gamma): g amma --> S-3 is given. A set {phi (gamma) \ gamma is an element of Gamma (G )} of embeddings is said to be realizable if there is an embedding f : G -- > S-3 such that the restriction map f/gamma is ambient isotopic to phi (gam ma) for any gamma is an element of Gamma (G). In this paper on seven specif ied graphs G, we give a necessary and sufficient condition for the set (phi (gamma) \ gamma is an element of Gamma (G)} to be realizable by using the second coefficients of Conway polynomials of knots.