Plane algebraic curves in fancy balls

TL;DR

This paper studies links in the 3-sphere that can be realized as intersections of algebraic curves with the boundary of a 4-ball, classifying such links with up to five crossings and exploring their properties.

Contribution

It introduces the concepts of strong and non-strong C-boundaries, provides examples of non-quasipositive strong C-boundaries, and classifies all C-boundaries with up to five crossings.

Findings

Some links are not C-boundaries.

All quasipositive links are strong C-boundaries.

Complete classification of C-boundaries with ≤5 crossings.

Abstract

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