Signature of Links

Let L be an oriented tame link in the three sphere S3. We study the Murasugi signature, o(L), and the nullity, tj(L). It is shown that o(L) is a locally flat topological concordance invariant and that tj(L) is a topological concordance invariant (no local flatness assumption here). Known results about the signature are re-proved (in some cases generalized) using branched coverings. 0. Introduction. Let L he an (oriented) tame link of multiplicity ju in the three-sphere S3. That is, L consists of p oriented circles Kx, . . . , K^ disjointly imbedded in S3. Various authors have investigated a numerical invariant, the signature of L (notation: o(L)). The signature was first defined for knots (ji = 1) by H. Trotter [21]. J. Milnor found another definition for this knot signature (see [12] ) in terms of the cohomology ring structure of the infinite cyclic cover of the knot complement. In [2], D. Erie showed that the definitions of Milnor and Trotter are equivalent. In [15], K. Murasugi formulated a definition of signature for arbitrary links. In this paper we investigate the Murasugi signature in the context of branched covering spaces. To be specific, let Z)4 denote the four dimensional ball with ó\D4 = S3, and let L C S3 be a link and F C Z)4 a properly imbedded, orientable, locally flat surface with bF L C S3. Let M denote the double branched cover of D4 along F. Then we show that o(L) is the signature of the four manifold M (see Lemma 1.1 and Theorem 3.1). Our proof of Theorem 3.1 contains the technicalities necessary to show this in the topological category. Using this viewpoint we are able to prove that o(L) is a topological concordance invariant (Theorem 3.8). We also rederive many of Murasugi's results, generalizing some of them (see Theorems 3.9-3.16). The paper is organized as follows: §1 contains the classical definitions of the signature and nullity of a link. It also deals with necessary background concerning branched coverings. Received by the editors November 4, 1974 and, in revised form, December 4, 1974 and April 11, 1975. AMS iMOS) subject classifications (1970). Primary 55A25.