I am currently working on the following paper by Lee Rudolph: <https://arxiv.org/abs/math/9307233>


Using Kronheimer-Mrowka's theorem, he proves in page 6 that the slice Euler characteristic of a given transverse $\mathbb C$-link $K\_f = V\_f \cap S^3$ with no singularities in $D^4$ is precisely the Euler characteristic of $V\_f \cap D^4$.


To that avail, he uses the fact that the link at infinity of the given complex algebraic curve $V\_f$ is isotopic to a torus knot $O(d,d)$ with d the degree of its projective completion, but I have been having a little trouble understanding why that is in the past few days.


I was hoping someone could help me get my head around this. I also have the understanding that the Milnor conjecture (or local Thom conjecture) is central in the given proof of the slice Bennequin inequality, but I am once again having trouble seeing where it is used.


Thanks and sorry if this is the wrong place to post this,


Paolo



Link at infinity of a complex algebraic curve transverse to S^3 and non-singular in D^4
I think one should be a bit cautious when speaking of *the* link at infinity in this context, and rather talk about the link at infinity *with respect to some line in $\mathbb{CP}^2$*.


Having said this, to get $T(d,d)$ as the link at infinity, I claim that it suffices to take a generic line in $\mathbb{CP}^2$, i.e. a line that intersects the projective completion of $V\_f$ transversely. (Note that this might well *not* be the line you use to compactify $\mathbb{C}^2$: for example, look at $f(x,y) = x^p-y^q$ with $p \neq q$.)


I can see two ways of justifying the claim. One is a deformation argument: you start from $f\_0(x,y) = x^d - y^d$, for which it is easy to check that the link at infinity (with respect to the line $z=0$) is $T(d,d)$, and then you deform from $f\_0$ to $f$ by moving the line at infinity together with the deformation, so that the intersections stay transverse at all times. This latter condition ensures that the link at infinity doesn't change (up to isotopy).


A different, more concrete way, is to view the link at infinity from the perspective of the line. Let me give things names: $C$ is going to be the projective completion of $V\_f$ and $L$ be a line that intersects $C$ transversely and that misses $B^4 \subset \mathbb{C}^2 \subset \mathbb{CP}^2$, that I will choose as the line at infinity. Choose also $N$ to be a small tubular neighbourhood of $L$: this is the disc bundle of Euler number $+1$ over $L$, which induces the *negative* Hopf fibration on the boundary. The link at infinity is, by definition, $C \cap -\partial N$ (note the change of orientation). Then $C \cap N$ is isotopic to the union of $d$ fibres of $N \to L$, so that the link at infinity is a union of $d$ fibres of the *positive* Hopf fibration. (The change from negative to positive Hopf fibration is due to the change in orientation of $\partial N$, and can be justified essentially by positivity of intersections.)


