The four genus of a link, Levine-Tristram signatures and satellites

TL;DR

This paper proves that Levine-Tristram signatures provide lower bounds for the 4-genus of links, extends a theorem on infection by string links, and constructs knots where these results determine the 4-genus.

Contribution

It offers a new proof relating signatures to the 4-genus and extends existing theorems to include infection by string links under certain conditions.

Findings

Levine-Tristram signatures bound the 4-genus from below.

Infection by string links does not increase the 4-genus under specific conditions.

Constructed knots where combined results determine the 4-genus.

Abstract

We give a new proof that the Levine-Tristram signatures of a link give lower bounds for the minimal sum of the genera of a collection of oriented, locally flat, disjointly embedded surfaces that the link can bound in the 4-ball. We call this minimal sum the 4-genus of the link. We also extend a theorem of Cochran, Friedl and Teichner to show that the 4-genus of a link does not increase under infection by a string link, which is a generalised satellite construction, provided that certain homotopy triviality conditions hold on the axis curves, and that enough Milnor's invariants of the infection string link vanish. We construct knots for which the combination of the two results determines the 4-genus.