k, there is an k-ary function which defines invariants according to which k of the links one selects, in which order. Milnor's invariants can be related to Massey products on the link complement (the complement of the link); this was suggested in , and made precise in and . As with Massey products, the Milnor invariants of length k + 1 are defined if all Milnor invariants of length less than or equal to k vanish. The first (2-fold) Milnor invariant is simply the linking number (just as the 2-fold Massey product is the cup product, which is dual to intersection), while the 3-fold Milnor invariant measures whether 3 pairwise unlinked circles are Borromean rings, and if so, in some sense, how many times (that is to say, the Borromean rings have a Milnor 3-fold invariant of 1 or –1, depending on order, but other 3-element links can have an invariant of 2 or more, just as linking numbers can be greater than 1). Another definition is the following: consider a link . Suppose that for and . Pick any Seifert surfaces for the respective link components, say, , such that for all . Then the Milnor 3-fold invariant equals minus the number of intersection points in counting with signs; . Milnor invariants can also be defined if the lower order invariants do not vanish, but then there is an indeterminacy, which depends on the values of the lower order invariants. This indeterminacy can be understood geometrically as the indeterminacy in expressing a link as a closed string link, as discussed below (it can also be seen algebraically as the indeterminacy of Massey products if lower order Massey products do not vanish). Milnor invariants can be considered as invariants of string links, in which case they are universally defined, and the indeterminacy of the Milnor invariant of a link is precisely due to the multiple ways that a given links can be cut into a string link; this allows the classification of links up to link homotopy, as in . Viewed from this point of view, Milnor invariants are finite type invariants, and in fact they (and their products) are the only rational finite type concordance invariants of string links; . The number of linearly independent Milnor invariants of length for m-component links is , where is the number of basic commutators of length k in the free Lie algebra