Some properties of simple minimal knots

TL;DR

This paper studies simple minimal knots, a class of knots derived from minimal disks in four-dimensional space, providing formulas for their braid representations and topological properties, and showing their relation to torus knots.

Contribution

The paper introduces a formula for the writhe number of simple minimal knots and explores their topological properties, including periodicity when parameters are not coprime.

Findings

Derived a formula for the writhe number of simple minimal knots.

Showed that simple minimal knots are periodic if parameters are not coprime.

Established that simple minimal knots generalize torus knots.

Abstract

A minimal knot is the intersection of a topologically embedded branched minimal disk in $\mathbb{R}^4$ $\mathbb{C}^2 $ with a small sphere centered at the branch point. When the lowest order terms in each coordinate component of the embedding of the disk in $\mathbb{C}^2$ are enough to determine the knot type, we talk of a simple minimal knot. Such a knot is given by three integers $N < p,q$; denoted by $K(N,p,q)$, it can be parametrized in the cylinder as $e^{i\theta}\mapsto (e^{Ni\theta},\sin q\theta,\cos p\theta)$. From this expression stems a natural representation of $K(N,p,q)$ as an $N$-braid. In this paper, we give a formula for its writhe number, i.e. the signed number of crossing points of this braid and derive topological consequences. We also show that if $q$ and $p$ are not mutually prime, $K(N,p,q)$ is periodic. Simple minimal knots are a generalization of torus knots.