Link group

In knot theory, an area of mathematics, the link group of a link is an analog of the knot group of a knot. They were described by John Milnor in his Ph.D. thesis, . Notably, the link group is not in general the fundamental group of the link complement.

In knot theory, an area of mathematics, the link group of a link is an analog of the knot group of a knot. They were described by John Milnor in his Ph.D. thesis, . Notably, the link group is not in general the fundamental group of the link complement. Another definition is the following: consider a link L=L_1\cup L_2\cup L_3 . Suppose that {\rm lk}(L_i,L_j)=0 for i,j=1,2,3 and i . Pick any Seifert surfaces for the respective link components, say, F_1,F_2,F_3 , such that F_i\cap L_j=\empty for all i\ne j . Then the Milnor 3-fold invariant equals minus the number of intersection points in F_1\cap F_2\cap F_3 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 k+1 for m-component links is m N_k - N_{k+1} , where N_k is the number of basic commutators of length k in the free Lie algebra on m generators, namely: N_k = \frac{1}{k}\sum_{d|m}\phi(d)\left(m^{k/d}\right) , where \phi is the Möbius function; see for instance . This number grows on the order of m^{k+1}/k^2 .

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