Generalization of the Conway-Gordon theorem and intrinsic linking on complete graphs (Q2033480)

This paper proves a theorem about two-component links that contain all vertices of an embedding of complete graphs: It is shown that for every spatial complete graph whose number of vertices is greater than six, the sum of the square of the linking numbers over all of the two-component Hamiltonian links (i.e. links containing all vertices) is determined explicitly in terms of the sum over all of the triangle-triangle constituent links. Corollaries for graphs with large number of vertices are proven and consequences for rectilinear (i.e. piecewise linear) embeddings of complete graphs are presented.