THE ARF AND SATO LINK CONCORDANCE INVARIANTS

The Kervaire-Arf invariant is a Z/2 valued concordance invariant of knots and proper links. The beta invariant (or Sato's invariant) is a Z valued concordance invariant of two component links of linking number zero discovered by J. Levine and studied by Sato, Cochran, and Daniel Ruberman. Cochran has found a sequence of invariants {beta-i} associated with a two component link of linking number zero where each beta-i is a Z valued concordance invariant and beta-0 = beta. In this paper we demonstrate a formula for the Arf invariant of a two component link L = X union Y of linking number zero in terms of the beta invariant of the link: arf(X union Y) = arf(X) + arf(Y) + beta-(X union Y) (mod 2). This leads to the result that the Arf invariant of a link of linking number zero is independent of the orientation of the link's components. We then establish a formula for \beta\ in terms of the link's Alexander polynomial DELTA-(x, y) = (x - 1)(y - 1) closed-integral(x, y): \beta-(L)\ = \closed-integral(1, 1)\. Finally we find a relationship between the beta-i invariants and linking numbers of lifts of X and Y in a Z/2 cover of the compliment of X union Y.