Singularity Knots of Minimal Surfaces in ${\mathbb{R}}^4$

Abstract

We study knots in ${\mathbb{S}}^3$ obtained by the intersection of a minimal surface in ${\mathbb{R}}^4$ with a small 3-sphere centered at a branch point. We construct examples of new minimal knots. In particular we show the existence of non-fibered minimal knots. We show that simple minimal knots are either reversible or fully amphicheiral; this yields an obstruction for a given knot to be an iterated knot of a minimal surface. Properties and invariants of these knots such as the algebraic crossing number of a braid representative and the Alexander polynomial are studied.