After the works of Kauffman-Banchoff and Yamasaki, it is known that a local move called the pass move is strongly related to the Arf invariant, which is equivalent to the parity of the coefficient of the degree two term in the Conway polynomial. Our main result is the following: There exists a pair of knots such that their Conway polynomials coincide, and that the sets of Conway polynomials of knots obtained from them by a single pass move do not coincide.
"The Gordian complex with pass moves is not homogeneous with respect to Conway polynomials." Hiroshima Math. J. 39 (3) 443 - 450, November 2009. https://doi.org/10.32917/hmj/1257544216