Studying links via closed braids. VI: A non-finiteness theorem
type: Item
instance of: scholarly article
title: Studying links via closed braids. VI: A non-finiteness theorem
zbMATH Open document ID: 0739.57002
author: Joan Birman
author: W. Menasco
published in: Pacific Journal of Mathematics
publication date: 1992-01-01T00:00:00Z
review text: Exchange moves take one closed \(n\)-braid representative of a link to another closed \(n\)-braid representative of the link, in general changing the conjugacy class. They can lead to examples where there are infinitely many conjugacy classes of \(n\)-braids representing a single link type. Two conjugacy classes in the braid group \(B_ n\) are ``exchange-equivalent'' if they have representatives which are equivalent under a finite series of exchange moves. Theorem: Let \({\mathcal L}\) be a link type which has infinitely many conjugacy classes of closed \(n\)-braid representatives. Then \(n\geq 4\) and the infinitely many conjugacy classes divide into finitely many exchange equivalence classes. This theorem is the last of the preliminary steps in the authors' program for the development of a calculus on links in \(S^ 3\).
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 4216262
zbMATH Keywords: Exchange moves
zbMATH Keywords: closed \(n\)-braid representative of a link
zbMATH Keywords: infinitely many conjugacy classes of \(n\)-braids representing a single link type
MaRDI profile type: Publication
full work available at URL: pjm.1992.156.265
OpenAlex ID: W2164128082
DOI: 10.2140/PJM.1992.156.265
description: scientific article; zbMATH DE number 4216262