On Milnor’s invariant for links

Abstract

Two links / and /' of multiplicity n and n' in R3 are said to be of the same type if there exists an orientation preserving homeomorphism of R3 onto itself that maps/ onto /' (and hence necessarily n = n'). A link is polygonal if it has a polygonal representative. Further, two links / and I' are said to be isotopic if there exists a continuous family h„ Oz^t z^l, of homeomorphisms of C„ into R3 with h0(C„) = I and hy(C„) = I'. These two concepts are not necessarily equivalent. In fact, the group of a link 1, G = ny(R3 — I), is an invariant of the link type but is not an isotopy invariant of the link. In 1952, K.T. Chen investigated the lower central series {Gq} of the group G of a polygonal link and proved that G/Gq is an isotopy invariant for all q [1]. (Gq is defined inductively by G. = G, Gi+1 = [G,Gf\, i = I, 2,-■-, where [G,Gi] is the subgroup generated by all aba~lb~x with aeG, beG,.) Later J. Milnor generalized this result for the group %'= ity(Jl — l), where Jl is an arbitrary open orientable 3-manifold and / is a fink in J( which is not necessarily polygonal [6]. In particular, for a link in 3-space R3, Milnor defined a numerical invariant p(iy ••• ik), where iy- ik is a sequence of positive integers between 1 and n. p will be called Milnor's invariant in this paper. p(ij), i¥=j, is the linking number of the ?th component I, and jth component lj of I. On the other hand, R.H. Fox defined the polynomial A(xjv,xJ with integral coefficients for a given link / of multiplicity n based on a presentation of the group G of / by means of his own free differential calculus [2], [3]. It is now called the Alexander polynomial of /. This is the natural generalization of the so called Alexander polynomial of a knot, (a knot being a link of multiplicity one). As is well known, the Alexander polynomial A(xi,Xj) of a link I, U lj of multiplicity two evaluated at x, = xi = 1 coincides up to sign with the linking number of I, and lj [7]. Therefore, we can write |A(x,,X;)X(=,i=1[ = |;i(i/)|.