Abstract
By the works of Levine [2] and Rolfsen [5], [6], it is known that a local move called a crossing-change is strongly related to the Alexander invariant. In this note, we will consider to what degree the relationship is strong. Let K be a knot, and $K^{\times}$ the set of knots obtained from a knot K by a single crossing-change. Let MK be the Alexander invariant of a knot K, and MK the set of the Alexander invariants $\{MK\}_{K{\in}\mathcal{K}}$ for a set of knots $\mathcal{K}$. Our main result is the following: If both $K_1$ and $K_2$ are knots with unknotting number one, then $MK_1=MK_2$ implies $MK_1^{\times}=MK_2^{\times}$. On the other hand, there exists a pair of knots $K_1$ and $K_2$ such that $MK_1=MK_2$ and $MK_1^{\times}{\neq}MK_2^{\times}$. In other words, the Gordian complex is not homogeneous with respect to Alexander invariants.