On C_n-moves for links

A C_n-move is a local move on links defined by Habiro and Goussarov, which can be regarded as a `higher order crossing change'. We use Milnor invariants with repeating indices to provide several classification results for links up to C_n-moves, under certain restrictions. Namely, we give a classification up to C_4-moves of 2-component links, 3-component Brunnian links and n-component C_3-trivial links, and we classify n-component link-homotopically trivial Brunnian links up to C_{n+1}-moves.