  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.
