The link group of an n-component link is essentially the set of -component links extending this link, up to link homotopy. In other words, each component of the extended link is allowed to move through regular homotopy, knotting or unknotting itself, but is not allowed to move through other component. This is a weaker condition than isotopy: for example, the Whitehead link has linking number 0, and thus is link homotopic to the unlink, but it is not isotopic to the unlink. The link group is not the fundamental group of the link complement, since the components of the link are allowed to move through themselves, though not each other, but thus is a quotient group of the link complement's fundamental group, since one can start with elements of the fundamental group, and then by knotting or unknotting components, some of these elements may become equivalent to each other.
Examples
The link group of the n-component unlink is the free group on n generators,, as the link group of a single link is the knot group of the unknot, which is the integers, and the link group of an unlinked union is the free product of the link groups of the components. is The link group of the Hopf link, the simplest non-trivial link – two circles, linked once – is the free abelian group on two generators, Note that the link group of two unlinked circles is the freenonabelian group on two generators, of which the free abelian group on two generators is a quotient. In this case the link group is the fundamental group of the link complement, as the link complement deformation retracts onto a torus. The Whitehead link is link homotopic to the unlink – though it is not isotopic to the unlink – and thus has link group the free group on two generators.
Milnor invariants
Milnor defined invariants of a link in, using the character which have thus come to be called "Milnor's μ-bar invariants", or simply the "Milnor invariants". For each 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 ; 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 tok vanish. The first Milnor invariant is simply the linking number, while the 3-fold Milnor invariant measures whether 3 pairwise unlinked circles are Borromean rings, and if so, in some sense, how many times. 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 equalsminus 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. 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 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 on m generators, namely: where is the Möbius function; see for instance. This number grows on the order of.
