Plane algebraic curves in fancy balls

Abstract

Boileau and Rudolph called a link $L$ in the $3$-sphere a $\mathbf{C}$-boundary if it can be realized as the intersection of an algebraic curve $A$ in ${\mathbf{C}}^2$ with the boundary of a smooth embedded $4$-ball $B$. They showed that some links are not $\mathbf{C}$-boundaries. We say that $L$ is a strong $\mathbf{C}$-boundary if $A\setminus B$ is connected. In particular, all quasipositive links are strong $\mathbf{C}$-boundaries. In this paper we give examples of non-quasipositive strong $\mathbf{C}$-boundaries and non-strong $\mathbf{C}$-boundaries. We give a complete classification of (strong) $\mathbf{C}$-boundaries with at most 5 crossings.

13 pages