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We give a new proof that the Levine-Tristram signatures of a link give lowerbounds for the minimal sum of the genera of a collection of oriented, locallyflat, disjointly embedded surfaces that the link can bound in the 4-ball. Wecall this minimal sum the 4-genus of the link. We also extend a theorem ofCochran, Friedl and Teichner to show that the 4-genus of a link does notincrease under infection by a string link, which is a generalised satelliteconstruction, provided that certain homotopy triviality conditions hold on theaxis curves, and that enough Milnor's invariants of the infection string linkvanish. We construct knots for which the combination of the two resultsdetermines the 4-genus.

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