Balanced algebraic unknotting, linking forms, and surfaces in three- and four-space (Q6562507)

Given a knot in the \(3\)-sphere, one can consider the locally-flat oriented sufaces embedded in the \(4\)-ball and bounded by the knot. Among these, the authors consider the ones whose complement has infinite cyclic fundamental group, which they call \textit{\(\mathbb{Z}\)-slice surfaces}. The \textit{\(\mathbb{Z}\)-slice genus} of a knot is the minimal genus of its \(\mathbb{Z}\)-slice surfaces. \par The main result of the paper is a characterisation of the existence of a \(\mathbb{Z}\)-slice surface of genus \(g\) in terms of \(3\)-dimensional properties. More precisely, the authors show that the existence of such a surface for a knot \(K\) is equivalent to: (1) the existence of an oriented surface in the \(3\)-sphere of genus \(g\) and with two boundary components, one \(K\) and one a knot with trivial Alexander polynomial; (2) the fact that \(K\) can be turned into a knot with trivial Alexander polynomial by changing \(g\) positive and \(g\) negative crossings; (3) the Blanchfield pairing of \(K\) can be presented by a Hermitian matrix of size \(2g\) with coefficients in \({\mathbb{Z}}[t^{\pm1}]\), such that the symmetric matrix obtained when specialising \(t=1\) has signature \(0\). The relationship between the Blanchfield pairing in the last characterisation and the linking paring of the double branched cover of the knot are exploited to provide lower bounds on the \(\mathbb{Z}\)-slice genus of the knot.\N\NFor \(g=0\) the result was already known as an application of Freedman's disk embedding theorem and part of the proof extends the argument of the \(g=0\) case.\N\NAs a consequence of their result, the authors show that the \(\mathbb{Z}\)-slice genus coincides with other notions of genus, notably the \textit{algebraic genus} (introduced by the authors in a previous paper [\textit{P. Feller} and \textit{L. Lewark}, Sel. Math., New Ser. 24, No. 5, 4885--4916 (2018; Zbl 1404.57008)]) and the the \textit{topological superslice genus}.\N\NA generalised version of the main theoem for knots embedded in integral homology spheres is also given. This more general version implies that the \(\mathbb{Z}\)-slice genus of a knot \(K\) coincides with the smallest number \(c\) such that \(K\) admits a \(\mathbb{Z}\)-disk in \(B^4\sharp({\mathbb{C}}P^2\sharp \overline{{\mathbb{C}}P^2})^{\sharp c}\).\N\NAs an application, the authors complete the determination of the \(\mathbb{Z}\)-slice genus for all prime knots with crossing numbers at most twelve.