topological "milnor's conjecture" on torus knots.

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

I realize the following answer comes somewhat belated.

Rudolph worked on this (Some topologically locally-flat surfaces in the complex projective plane), and more recently Baader, Feller, Liechti and myself (https://arxiv.org/abs/1509.07634).

Here is a brief summary of what is known:

Levine-Tristram signatures give a lower bound for the topological 4-genus of torus knots, which is asymptotically equal to half of their 3-genus. This is the best we have; other methods, like Casson-Gordon invariants or $L^2$-signatures, might give better bounds, but they have not been computed for torus knots.

Upper bounds may be obtained by the "algebraic genus": constructing locally flat slice surface by ambient surgery, using Freedman's result that Alexander-polynomial-1 knots are topologically slice. In this way, one can prove that for small (3-genus $\leq$ 14) torus knots, the topological 4-genus equals the maximum of the Levine-Tristram signatures. In particular, for the examples Ian mentions, one has $g_4^t(T(3,7))=g_4^t(T(4,5))=5$. More generally, for $p<q$ one has $g_4^t(T(p,q)) < g_3(T(p,q))$ unless $p=2$ or $(p,q)\in\{(3,4),(3,5)\}$.

With the same method, one finds that asymptotically, the topological 4-genus of large torus knots is at most three quarters of their 3-genus.

Feeling adventurous, one could conjecture that for all torus knots, the topological 4-genus equals the maximum of the Levine-Tristram signatures.

Answer 3

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.