  An oriented link is called $\mathbb C$-boundary if it is realizable as
$(\partial B,A\cap\partial B)$ 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.
