Cross-sections of unknotted ribbon disks and algebraic curves

TL;DR

This paper demonstrates that many complex links can be realized as cross-sections of unknotted holomorphic disks in four-dimensional space, advancing understanding of ribbon surfaces and their properties.

Contribution

It provides methods to produce unknotted ribbon surfaces with specific cross-sections, including unknotted Lagrangian disks with nontrivial links, solving parts of a known problem.

Findings

Many nontrivial links arise as cross-sections of unknotted holomorphic disks.

Techniques to produce unknotted ribbon surfaces with prescribed cross-sections.

Construction of unknotted Lagrangian disks with nontrivial links.

Abstract

We resolve parts (A) and (B) of Problem 1.100 from Kirby's list by showing that many nontrivial links arise as cross-sections of unknotted holomorphic disks in the four-ball. The techniques can be used to produce unknotted ribbon surfaces with prescribed cross-sections, including unknotted Lagrangian disks with nontrivial cross-sections.