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.



topological "milnor's conjecture" on torus knots.
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.


