Analytic knots, satellites and the 4-ball genus

Call a knot in the unit sphere in complex affine 2-space
analytic (respectively, smoothly analytic) if it bounds a complex curve
(respectively, smooth complex curve) in the complex ball. Let $K$ be
a smoothly analytic knot. We show that  there is a tubular neighbourhood
of $K$ with the following properties. There is a sharp lower bound  of the
4-ball genus of an arbitrary analytic knot $L$ contained in it in terms of
the 4-ball genus of $K$ and the "Umlaufzahl" of $L$ with respect to $K$.
Moreover, analytic links in this tubular neighbourhood can be described.
This is related to a variety of (partly open) mathematical questions
concerning branched coverings $p:Y \to X$ of (smoothly bounded) open Riemann
surfaces $X$, embeddings of $Y$ into a disc bundle over $X$ and the braid
formed by the related embedding of the boundary of $Y$.