Some properties of simple minimal 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.
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Submitted on : Monday, December 10, 2012 - 10:56:04 PM
Last modification on : Wednesday, December 4, 2019 - 10:51:11 AM
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  • HAL Id : hal-00763478, version 1
  • ARXIV : 1212.2347

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Marina Ville, Marc Soret. Some properties of simple minimal knots. 2012. ⟨hal-00763478⟩

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