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

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

2007-02-09

arxiv.org/abs/math/0702254v1

Mathematics

Differential Geometry

Scientific paper

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.

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