On Conway-Gordon type theorems for graphs in the Petersen family (Q2850717)

If \(f:G\rightarrow S^{3}\) is an imbedding of a graph \(G\) in the three-sphere then the image of each cycle of \(G\) is a knot in \(f(G)\) and the image of each pair of vertex-disjoint cycles of \(G\) is a two-component link in \(f(G)\). \textit{J. H. Conway} and \textit{C. McA. Gordon} [J. Graph Theory 7, 445--453 (1983; Zbl 0524.05028)] showed that every imbedding of the complete graph \(K_{6}\) in the three-sphere contains a non-trivial link, and every imbedding of \(K_{7}\) in the three-sphere contains a non-trivial knot. The first result extends directly to graphs in the ``Petersen family,'' i.e., graphs obtainable from \(K_{6}\) by \(\triangle Y\)- and \(Y \triangle\)-exchanges. Later, \textit{N. Robertson} et al. [J. Comb. Theory, Ser. B 64, No. 2, 185--227 (1995; Zbl 0832.05032)] verified a conjecture of \textit{H. Sachs} [in: Finite and infinite sets, 6th Hung. Combin. Colloq., Eger/Hung. 1981, Vol. II, Colloq. Math. Soc. János Bolyai 37, 649--662 (1984; Zbl 0568.05026)] that the Petersen family graphs are the minor-minimal graphs with the property that every imbedding in the three-sphere contains a non-trivial link.NEWLINENEWLINEIn the paper under review, the authors give a formula that relates the linking numbers of two-component links and the Conway polynomials of knots in imbedded Peterson family graphs. This formula lifts a congruence of Conway and Gordon from \(\mathbb Z/2\mathbb Z\) to \(\mathbb Z\). In addition, it provides a new proof of an unpublished result of \textit{O'Donnoll} [``Knotting and linking in the Petersen family'', \url{arxiv:1008.0377}] that certain combinations of linking numbers in imbedded Petersen family graphs guarantee the presence of knotted cycles.