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

Abstract : 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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https://hal.archives-ouvertes.fr/hal-00360793
Contributor : Marc Soret <>
Submitted on : Thursday, February 12, 2009 - 1:07:43 AM
Last modification on : Friday, December 6, 2019 - 1:35:55 AM

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Marc Soret, Marina Ville. Singularity Knots of Minimal Surfaces in $\mathbb{R}^4$. Journal of Knot Theory and Its Ramifications, World Scientific Publishing, 2011, 20 (4), pp.513-546. ⟨10.1142/S0218216511009406⟩. ⟨hal-00360793⟩

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