Shake genus and slice genus

Abstract

An important difference between high-dimensional smooth manifolds and smooth $4$–manifolds that in a $4$–manifold it is not always possible to represent every middle-dimensional homology class with a smoothly embedded sphere. This is true even among the simplest $4$–manifolds: $X_0\left(K\right)$ obtained by attaching an $0$–framed $2$–handle to the $4$–ball along a knot $K$ in $S^3$. The $0$–shake genus of $K$ records the minimal genus among all smooth embedded surfaces representing a generator of the second homology of $X_0\left(K\right)$ and is clearly bounded above by the slice genus of $K$. We prove that slice genus is not an invariant of $X_0\left(K\right)$, and thereby provide infinitely many examples of knots with $0$–shake genus strictly less than slice genus. This resolves Problem 1.41 of Kirby’s 1997 problem list. As corollaries we show that Rasmussen’s $s$ invariant is not a $0$–trace invariant and we give examples, via the satellite operation, of bijective maps on the smooth concordance group which fix the identity but do not preserve slice genus. These corollaries resolve some questions from a conference at the Max Planck Institute, Bonn (2016).