Local Moves and Gordian Complexes, II

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× the set of knots obtained from a knot K by a singl crossing-change. Let MK be the Alexander invariant of a knot K, and MK the set of the Alexander invariants {MK}K2K for a set of knots K. Our main result is the following: If both K1 and K2 are knots with unknotting number one, then MK1 = MK2 implies MK×1 = MK×2 . On the other hand, there exists a pair of knots K1 and K2 such that MK1 = MK2 and MK×1 6= MK×2 . In other words, the Gordian complex is not homogeneous with respect to Alexander invariant