Shake genus and slice genus

TL;DR

This paper investigates the difference between the shake genus and slice genus of knots, proving that slice genus is not an invariant of certain 4-manifolds and providing examples where the shake genus is strictly less.

Contribution

It demonstrates that slice genus is not an invariant of the 4-manifold $X_0(K)$, offering new examples and resolving longstanding problems in knot theory and 4-manifold topology.

Findings

Slice genus is not an invariant of $X_0(K)$.

Provided infinitely many knots with shake genus less than slice genus.

Showed Rasmussen's s invariant is not a $0$-trace invariant.

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: $X_0(K)$ 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 $X_0(K)$ and is clearly bounded above by the slice genus of $K$. We prove that slice genus is not an invariant of $X_0(K)$, and thereby provide infinitely many examples of knots with $0$-shake genus strictly less than slice genus. This resolves Problem 1.41 of [Kir97]. 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 [4MKC16].