The Gordian complex with pass moves is not homogeneous with respect to Conway polynomials (Q845255)

A pass move between knot diagrams is a change of a set of four crossings which may be viewed as a change of crossings of a pair of oriented bands. These moves were introduced by \textit{L. Kauffman} in [Formal knot theory. Mathematical Notes, 30. Princeton, New Jersey: Princeton University Press (1983; Zbl 0537.57002)] and were shown to preserve Arf invariants (i.e. the mod 2 coefficient of \(z^2\) in the Conway polynomial) by \textit{L. Kauffman} and \textit{T. F. Banchoff} [Am. Math. Mon. 84, 168--185 (1977; Zbl 0362.57001)] and independently by Yamasaki [``On a surface in \(S^3\)'', Surikaisekikenkyusho Kokyuroku 297, 92--99 (1977)]. In the paper reviewed here, the authors study the collection of knots differing from a given knot by a single pass move. They show that there exist knots \(K_1\) and \(K_2\) with identical Conway polynomials for which the set of all Conway polynomials of knots differing from \(K_1\) by a single pass move is not identical to the set of all Conway polynomials of knots differing from \(K_2\) by a single pass move.