A geometric formula for Haefliger knots
type: Item
instance of: scholarly article
title: A geometric formula for Haefliger knots
zbMATH Open document ID: 1060.57021
author: Masamichi Takase
published in: Topology
publication date: 2004-11-23T00:00:00Z
review text: Haefliger showed that the group of isotopy classes of smooth \((4k-1)\)-knots in \(S^{6k}\) is infinite cyclic, and gave a simple, explicit construction of a generator. He defined an invariant \(\Omega(F)\in\mathbb{Z}\) determining the isomorphism in terms of the cohomology of a submanifold of \(S^{6k}\) with boundary \(F\). This paper gives a new formula for \(\Omega(F)\), using immersions rather than embeddings. Haefliger's work is used to show that every such knot \(F\) extends to a generic immersion \(\widetilde{F}\) of a compact orientable \(4k\)-manifold \(V\) with boundary \(S^{4k-1}\). The new formulation of the invariant involves double and triple point data of \(\widetilde{F}\), the \(k\)th normal Pontrjagin number of \(V\cup{D^{4k}}\) and a Hopf invariant. The linking number of \textit{A. Szűcs} [Comment. Math. Helv. 61, 360-369 (1986; Zbl 0602.57021)], interpreted via the Hopf invariant, plays a key role.
Mathematics Subject Classification ID: 57R40
Mathematics Subject Classification ID: 57R45
Mathematics Subject Classification ID: 57R52
Mathematics Subject Classification ID: 57Q45
zbMATH DE Number: 2117936
zbMATH Keywords: Haefliger knot
zbMATH Keywords: Hopf invariant
zbMATH Keywords: smooth isotopy
zbMATH Keywords: Szűcs' linking
reviewed by: Jonathan A. Hillman
MaRDI profile type: Publication
full work available at URL: j.top.2004.03.001
OpenAlex ID: W2108627543
DOI: 10.1016/J.TOP.2004.03.001
description: scientific article; zbMATH DE number 2117936