Constraints on Alexander polynomials of certain two-component links (Q1886715)

This paper considers 2-component links which are constructed in the following manner. Take a closed \(2m\)-braid \(B\) which represents the trivial knot and consider the double branched cover \(D\) of \(S^3\) branched along \(B\). The axis of \(B\) will lift to a 2-component link \(L_B\) in \(D\) (note that \(D\) is diffeomorphic to \(S^3\)). This construction was used by Fintushel-Stern to exhibit different symplectic surfaces representing the same homology class in a complex surface. These surfaces were constructed from the braids \(B\) and they were distinguished by looking at the Seiberg-Witten invariants of covers of the complex surface, branched along the surfaces. A direct relation between these SW-invariants and the Alexander polynomial of \(L_B\) enabled their calculation. In the present paper the Fintushel-Stern argument is reversed, and the relation between SW-invariants and Alexander polynomials is used to exhibit a family of mod 2 congruences satisfied by the Alexander polynomial of \(L_B\). These congruences are generalized to links which are lifts of any knot homologous to the axis in the complement of \(B\).