A refinement of the Conway–Gordon theorems

In 1983, Conway–Gordon showed that for every spatial complete graph on 6 vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on 7 vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruent to 1 modulo 2. In this article, we give integral lifts of the Conway–Gordon theorems above in terms of the square of the linking number and the second coefficient of the Conway polynomial. As applications, we give alternative topological proofs of theorems of Brown–Ramírez Alfonsín and Huh–Jeon for rectilinear spatial complete graphs which were proved by computational and combinatorial methods.