The mathbb Z-genus of boundary links

Abstract

The mathbb 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 mathbb 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-shake genus, equals the mathbb Z-genus of the knot.