Plane algebraic curves in fancy balls

Boileau and Rudolph [1] called an oriented link $L$ in the 3-sphere a \textit{$\mathbb{C}$-boundary} if it can be realized as the intersection of an algebraic curve $A$ in $\mathbb{C}^2$ and the boundary of a smooth embedded closed 4-ball $B$. They showed that some links are not $\mathbb{C}$-boundaries. We say that $L$ is a \textit{strong $\mathbb{C}$-boundary} if $A\setminus B$ is connected. In particular, all quasipositive links are strong $\mathbb{C}$-boundaries.
In this paper we give examples of non-quasipositive strong $\mathbb{C}$-boundaries and non-strong $\mathbb{C}$-boundaries. We give a complete classification of (strong) $\mathbb{C}$-boundaries with at most five crossings.