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

TL;DR

This paper introduces the concept of the $\\mathbb{Z}$-genus for boundary links, characterizes it via Blanchfield forms, and relates it to the shake genus, providing new insights into link and knot topology.

Contribution

It characterizes the $\mathbb{Z}$-genus of boundary links using Blanchfield forms and establishes its equality with the $\mathbb{Z}$-shake genus for knots.

Findings

Characterization of $\mathbb{Z}$-genus via Blanchfield forms.

Equivalence of $\mathbb{Z}$-shake genus and $\mathbb{Z}$-genus for knots.

Applications to boundary link topology.

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.