Local Moves and Gordian Complexes, II
نویسنده
چکیده
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 single crossing-change. Let MK be the Alexander invariant of a knot K, and MK the set of the Alexander invariants {MK}K∈K 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 invariants.
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