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

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 generalized satellite construction, provided that certain homotopy triviality conditions hold on the axis curves, and that enough Milnor's μ¯¯¯μ¯-invariants of the closure of the infection string link vanish. We construct knots for which the combination of the two results determines the 4-genus