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',
and which gives a complete topological characterization of
Goussarov-Vassiliev (or finite type) knot invariants: two knots cannot
be distinguished by finite type invariants of degree $<n$ if and
only if they are related by a finite sequence of $C_n$-moves.
The analogous statement is known to be false for links in general, but
it is conjecturally true for string links, which are certain links with
This conjecture is partly supported by the fact that Milnor invariants,
which are invariants (of both links and string links) generalizing the
linking number, are of finite type only for string links.
In this talk, we will classify $l$-component string links up to
$C_n$-move for $n\le 5$, by explicitly giving complete sets of low
degree finite type invariants. In addition to Milnor invariants, these
include several `new' string link invariants constructed by evaluating
knot invariants on certain closure of (cabled) string links.
If time allows, we will also give similar results for concordance
finite type invariants.