Cn-MOVE AND ITS DUPLICATED MOVE OF LINKS

Abstract

A local move is a pair of tangles with same end points. Habiro defined a system of local moves, Cn-moves, and showed that two knots have the same Vassiliev invariants of order ≤ n - 1 if and only if they are transformed into each other by Cn-moves. We define a local move, βn-move, which is obtained from a Cn-move by duplicating a single pair of arcs with same end points. Then we immediately have that a Cn+1-move is realized by a βn-move and that a βn-move is realized by twice Cn-moves. In this note we study the relation between Cn-move and βn-move, and in particular, give answers to the following questions: (1) Is a βn-move realized by a finite sequence of Cn+1-moves? (2) Is Cn-move realized by a finite sequence of βn-moves?