$C_n$-moves and $V_n$-equivalence for Links

Abstract

K. Habiro defined a $C_n$-move which is a local move on oriented links. He also proved that two knots are not distinguished by any Vassiliev invariants of order less than $n$ if and only if they are related by a finite sequence of $C_n$-moves. In the case of $n \geq 3$, it is known that the result does not hold for links. In this note we will introduce a special $C_n$-move and give a ``geometrical'' necessary and sufficient condition using the terms of $C_n$-moves for that two links are not distinguished by any Vassiliev invariants of order less than $3$ or $4$.