Equivariant isotopy of unknots to round circles
type: Item
instance of: scholarly article
title: Equivariant isotopy of unknots to round circles
zbMATH Open document ID: 0830.53002
DOI: 10.1016/0166-8641(94)00086-I
author: Feng Luo
author: Michael Freedman
published in: Topology and its Applications
publication date: 1995-10-18T00:00:00Z
review text: The Möbius group \(\text{Möb} (\mathbb{S}^3)\) is the (10)-dimensional Lie group of two components generated by inversions in 2-sphere. The authors prove that for every compact subgroup \(G\) of \(\text{Möb}(\mathbb{S}^3)\), if \(\gamma \subset \mathbb{S}^3\) is a smooth unknotted \(G\)-invariant simple closed curve, then there exists a smooth family of \(G\)-invariant simple closed curves \(\gamma_t\), \(0 \leq t \leq 1\), with \(\gamma_0 = \gamma\) and \(\gamma_1\) a round circle. This interesting result links with the study of the existence of a Möbius invariant energy functional defined on the set of smooth simple closed curves in the 3-sphere (with values in the positive real numbers), attaining its minimum precisely on round circles.
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 53A30
zbMATH DE Number: 792531
zbMATH Keywords: knot
zbMATH Keywords: unknotting
zbMATH Keywords: Möbius group
zbMATH Keywords: Möbius invariant energy functional
zbMATH Keywords: round circles
reviewed by: Alberto Cavicchioli
MaRDI profile type: Publication
cites work: Q3733804
cites work: Difféotopies des espaces lenticulaires
cites work: Möbius energy of knots and unknots
cites work: A proof of the Smale conjecture, \(Diff(S^ 3)\simeq O(4)\)
cites work: The classical Plateau problem and the topology of three-dimensional manifolds. The embedding of the solution given by Douglas-Morrey and an analytic proof of Dehn's lemma
cites work: Energy of a knot
cites work: Knoten mit zwei Brücken
cites work: Q5628326
full work available at URL: 0166-8641(94)00086-i
OpenAlex ID: W2157947333
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description: scientific article; zbMATH DE number 792531