Desingularization of branch points of minimal surfaces in $\mathbb{R}^4$ (II)

TL;DR

This paper presents a method to desingularize branch points of minimal surfaces in four-dimensional space by creating immersions with only transverse double points, revealing braid representations of knots associated with the surfaces.

Contribution

It introduces a new desingularization technique for branch points in minimal surfaces in $\

Findings

Desingularization produces immersions with transverse double points.

Double points encode braid representations of knots.

Method applies to minimal disks in $\

Abstract

We desingularize a branch point $p$ of a minimal disk $F_0(\mathbb{D})$ in $\mathbb{R}^4$ through immersions $F_t$'s which have only transverse double points and are branched covers of the plane tangent to $F_0(\mathbb{D})$ at $p$. If $F_0$ is a topological embedding and thus defines a knot in a sphere/cylinder around the branch point, the data of the double points of the $F_t$'s give us a braid representation of this knot as a product of bands.