Plane algebraic curves in fancy balls

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