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

Abstract

The Z-genus of a link L in S 3 is the minimal genus of a locally flat, embedded, connected surface in D 4 whose boundary is L and with the fundamental group of the complement infinite cyclic. We characterise the 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 Z-shake genus, equals the Z-genus of the knot.