The total curvature of a knotted torus
type: Item
instance of: scholarly article
title: The total curvature of a knotted torus
zbMATH Open document ID: 0602.53038
author: Nicolaas H. Kuiper
author: William H. III Meeks
published in: Journal of Differential Geometry
publication date: 1987-01-01T00:00:00Z
review text: In this paper dedicated to James Eells jun., the authors generalize to torus embeddings in \({\mathbb{R}}^ 3\) the theorem of Milnor: The normalized total curvature of an honest knot \(\gamma\) is \(\pi (\gamma)=\int | \rho ds| \quad /\quad \pi >2 B[\gamma].\) The infimum of \(\tau\) in the isotopy class [\(\gamma\) ] is twice the bridge index B[\(\gamma\) ] of the knot. A torus T, embedded in \({\mathbb{R}}^ 3\subset {\mathbb{R}}^ 3\cup \{\infty \}=S^ 3\), divides \(S^ 3\) in two parts, one of which is a tube around a knot \(\gamma\). Theorem: The normalized total curvature \(\tau (T)=\int | K d\sigma | /2\pi\) of an honestly knotted torus \(T\subset {\mathbb{R}}^ 3\) has infimum 4 B[\(\gamma\) ]. This infimum cannot be attained: \(\tau (T)>4 B[\gamma]\). This paper uses results of an earlier paper [Invent. Math. 77, 25-69 (1984; Zbl 0553.53034)] where the authors proved attainment of an infimum for certain knotted surfaces of genus \(g\geq 3\).
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53A05
zbMATH DE Number: 3971671
zbMATH Keywords: total curvature
zbMATH Keywords: honest knot
zbMATH Keywords: knotted torus
zbMATH Keywords: knotted surfaces
Wikidata QID: Q101231904
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214441483
OpenAlex ID: W1502596804
DOI: 10.4310/JDG/1214441483
description: scientific article; zbMATH DE number 3971671