Milnor and Finite Type Invariants of Plat-closures

Finite type invariants of knots or links can be defined combinatorially using only link projections in S. In this setting it can be seen that every Jones-type polynomial invariant (quantum invariants) is equivalent to a sequence of finite type invariants. See [B2, BN] and references therein. Although Vassiliev’s original approach to finite type knot invariants ([V]) rests on topological foundations it doesn’t offer any insight into how the invariants are related to the topology of the complement of an individual knot or link. Milnor’s μ̄-invariants ([M1,2]) are integer link concordance invariants defined in terms of algebraic data extracted from the link group. It is known that the μ̄-invariants of length k detect exactly when the longitudes of the link lie in the k-th term of the lower central series of the link group. As was shown in [BN2] and [Li] (and further illustrated in [H-M]), these invariants can be thought of as the link homotopy (or link concordance) counterpart of the theory of finite type invariants. As it follows from the results in the above manuscripts if all finite type invariants of orders ≤ m vanish for a link L, then all its Milnor invariants of length ≤ m+ 1 vanish. In [K-L] we related the finite type invariants of a knot to some geometric properties of its complement. We showed that the invariants of orders ≤ m detect when a knot bounds a regular Seifert surface S, whose complement looks, modulo the first m+ 2 terms of the lower central series of its fundamental group, like the complement of a null-isotopy. In the present note we use the techniques of [K-L] to study finite type invariants of n-component links that admit an n-bridge presentation (see for example [Ro]). These links are precisely the ones realized as plat-closures of