Singularity Knots of Minimal Surfaces in ℝ⁴

2007/02/09 by Marc Soret, Soret, Marc, Marina Ville +1
Mathematics · #53C42 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.math/0702254

openalex publication_date 2007/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study knots in \mathbbS³ obtained by the intersection of a minimal surface in ℝ⁴ 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.

Related

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• Desingularization of branch points of minimal surfaces in ℝ⁴ (II)
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• Lipschitz geometry of surface germs in ℝ⁴: metric knots