C-boundary links up to six crossings

An oriented link is called $\mathbb{C}$-boundary if it is realizable as $(\mathrm{\partial }B,A\cap \mathrm{\partial }B)$ where $A$ is an algebraic curve in ${\mathbb{C}}^2$ and $B$ is an embedded $4$-ball. This notion was introduced by Michel Boileau and Lee Rudolph in 1995. In a recent joint paper with N.G. Kruzhilin we gave a complete classification of $\mathbb{C}$-boundaries with at most 5 crossings. In the present paper a more regular method of construction of $\mathbb{C}$-boundaries is proposed and the classification is extended up to 6 crossings.