On classifications of links up to C_n-moves

A C_n-move (nin{Bbb N}) is a local move on links defined by Habiro, which can be regarded as a 'higher order crossing change'. The C_n-equivalence is an equivalence relation on links generated by C_n-move. The C_m-equivalence implies the C_n-equivalence for m>n. So the C_n-classification, which is the classification up to C_n-equivalence, of links becomes finer as n increases. The C_2-classification of links and the C_3-classification of links with 2 or 3 components, or of algebraically split links are known. Here we give several classifications of certain sets of links by using Milnor invariants. More precisely, we give the following classification:
(1) C_n-classification of n-component Brunnian links.
(2) C_{n+1}-classification of n-component Brunnian links which are C_n-equivalent to trivial.
(3) C_4-classification of 2-component links, 3-component Brunnian links or n-component Brunnian links which are C_3-equivalent to trivial.