Question

Here's a question that has come up in a couple of talks that I have given recently.

The 'classical' way to show that there is a knot $K$ that is locally-flat slice in the 4-ball but not smoothly slice in the 4-ball is to do two things

1. Compute that the Alexander polynomial of $K$ is 1, and so by results of Freedman's you know that $K$ is locally-flat slice.

2. (due to Rudolph) Somehow obtain a special diagram of $K$ (or utilize a more subtle argument) to show that you can present $K$ as a separating curve on a minimal Seifert surface for a torus knot. Since we know (by various proofs, the first due to Kronheimer-Mrowka) that the genus of torus knot is equal to its smooth 4-ball genus (part of Milnor's conjecture), the smooth 4-ball genus of $K$ must be equal to the genus of the piece of the torus knot Seifert surface that it bounds, and this is $\geq 1$.

Boiling the approach of 2. down to braid diagrams, you come up with the slice-Bennequin inequality.

Well, here's the thing. I have this smooth cobordism from the torus knot to $K$, and then I know that $K$ bounds a locally-flat disc. This means that the locally-flat 4-ball genus of the torus knot must be less than its smooth 4-ball genus. So if you were to conjecture that the locally flat 4-ball genus of a torus knot agrees with its smooth 4-ball genus, you would be wrong.

My question is - are there any conjectures out there on the torus knot locally-flat genus? Even asymptotically? Any results? Any way known to try and study this?

Thanks, Andrew.

Answer 1

Related to the early investigation of the Thom conjecture,the G-signature thm was used circa 1970 to give 4-ball genus bounds for torus knots which asyptocically (in some cases) were a fixed fraction of what we now know to be the smooth category answer. I belive Larry Tayor observed (in the '70s or early 80s) that these G-signature bounds hold in the topologically flat world as well. Thus, I believe, there there are families of torus knots where the the flat-4-ball geunus is known to be at least some known fraction of the smooth 4-ball genus. Sorry I don't have the references at hand.

Answer 2

The signature/2 gives a lower bound on the 4-ball genus. Looking through a table of torus knots, the first ones I found where the smooth genus is > signature/2 were T(7,3), T(5,4). I don't remember the example from your talk, but can you show that these ones have smaller topological genus? It's possible that there are better lower estimates on the 4-genus coming from other sorts of signatures.