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

Abstract

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 µ-invariants of the infection string link vanish.We construct knots for which the combination of the two results determines the 4-genus.

Branched covers of quasi‐positive links and L‐spaces

“…(See [, Theorem 5.19], for instance.) A recent paper of Mark Powell provides a synopsis of the work that has been done on the Murasugi–Tristram inequality and proves it in the case that is locally flat with components [, Theorem 1.4]. As the locally flat case of the is key to our arguments, we include a self‐contained proof below.…”

Stably slice disks of links

“…The proof of Theorem 3.10 relies on the more technical statement provided by Theorem 3.8. This latter result is a generalization of [16,Theorem 3.7] which is itself a generalization of the Murasugi-Tristram inequality [11,20,21,24,35,38,44,46]. We state this result as it might be of independent interest, but refer to Definition 3.7 for the definition of a null-homologous cobordism.…”

“…The 4-dimensional interpretations of both the Levine-Tristram and multivariable signature were originally stated using finite branched covers and certain eigenspaces associated to their second homology group [11,45]; the use of twisted homology only emerged later [16,18,38,46]. We briefly recall the construction of the aforementioned eigenspaces in the one variable case.…”

On the topological 4-genus of torus knots

“…While well-known to experts, until recently this lower bound had not been explicitly stated in the literature in the topological setting (compare [Tri69] for the smooth setting). This gap in the literature was closed by Powell with a new proof [Pow16].…”