Some properties of simple minimal knots

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.