The $$\mathbb Z$$-genus of boundary links

The $$\mathbb Z$$ Z -genus of a link L in $$S^3$$ S 3 is the minimal genus of a locally flat, embedded, connected surface in $$D^4$$ D 4 whose boundary is L and with the fundamental group of the complement infinite cyclic. We characterise the $$\mathbb Z$$ Z -genus of boundary links in terms of their single variable Blanchfield forms, and we present some applications. In particular, we show that a variant of the shake genus of a knot, the $$\mathbb Z$$ Z -shake genus, equals the $$\mathbb Z$$ Z -genus of the knot.