
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: obtained by attaching an -framed 2-handle to the 4-ball along a knot in . The -shake genus of records the minimal genus among all smooth embedded surfaces representing a generator of the second homology of and is clearly bounded above by the slice genus of . We prove that slice genus is not an invariant of , and thereby provide infinitely many examples of knots with -shake genus strictly less than slice genus. This resolves Problem 1.41 of [Kir97]. As corollaries we show that Rasmussen's invariant is not a -trace invariant and we give examples, via the satellite…
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