Finite type invariants of string links and the HOMFLYPT polynomial of knots

A family of finite type invariants of string link is given by the HOMFLYPT polynomial of knots that are obtained by various closure operations on (cabled) string links.

In this talk we will show the following:
(1) These invariants, together with Milnor invariants of length $\le 5$, give classifications of $n$-string links up to $C_k$-equivalence for $k\leq 5$, and give a complete set of finite type invariants of degree $\leq 4$.
(2) Any Milnor invariant of length $n + 1(>2)$ of a $C_n$-trivial string link is expressed as a linear combination of such invariants.

(joint work with Jean-Baptiste Meilhan)