The Arf and Sato link concordance invariants

Abstract:

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 $L = X \cup Y$ of linking number zero in terms of the $\beta$ invariant of the link: \[ \operatorname {arf} (X \cup Y) = \operatorname {arf} (X) + \operatorname {arf} (Y) + \beta (X \cup Y)\quad (\bmod 2).\] 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 $\Delta (x,y) = (x - 1)(y - 1)f(x,y)$: \[ |\beta (L)| = |f(1,1)|.\] 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 $X \cup Y$.