Converses to Generalized Conway–Gordon Type Congruences

It is known that for every spatial complete graph on $n\ge 7$ vertices, the summation of the second coefficients of the Conway polynomials over the Hamiltonian knots is congruent to $r_{n}$ modulo $(n-5)!$, where $r_{n} = (n-5)!/2$ if $n=8k,8k+7$, and $0$ if $n\neq 8k,8k+7$. In particular the case of $n=7$ is famous as the Conway–Gordon $K_{7}$ theorem. In this paper, conversely, we show that every integer $(n-5)! q + r_{n}$ is realized as the summation of the second coefficients of the Conway polynomials over the Hamiltonian knots in some spatial complete graph on $n$ vertices.