Knots, braid index and dynamical type
type: Item
instance of: scholarly article
title: Knots, braid index and dynamical type
zbMATH Open document ID: 0828.57006
DOI: 10.1016/0040-9383(94)90014-0
author: Jérôme Los
published in: Topology
publication date: 1996-01-10T00:00:00Z
review text: We recall first a couple of well-known facts. Given a knot \(K\) in the 3- sphere \(S^3\), a theorem of Thurston says that \(K\) is either a torus knot, a satellite knot, or its complement \(S^3 - K\) admits a hyperbolic structure. In the latter case, the knot is said to be hyperbolic. The braid index set \(Bi (K)\) of \(K\) is the set of braids representing \(K\) with the minimum number of strands. A braid on \(n\) strands defines an isotopy class of homeomorphisms of the \(n\)-punctured disk. Homeomorphisms of surfaces are classified, by another result of Thurston, into three types: periodic, reducible and pseudo-Anosov. In this paper, the author studies the relation between the two results of Thurston and he proves the following: (i) \(K\) is a torus knot if and only if every braid in \(Bi (K)\) is periodic. (ii) \(K\) is a hyperbolic or a non-braided satellite knot if and only if every braid in \(Bi (K)\) is pseudo-Anosov; (iii) \(K\) is a braided satellite knot if and only if for every \(\beta\) in \(Bi (K)\), there exists a finite sequence of exchange moves (and conjugacies) in \(Bi (K)\): \(\beta \to \beta_1 \to \cdots \to \beta_N\) such that \(\beta_N\) is reducible. (Exchange moves are certain transformations which have been defined by Birman and Menasco.) Another natural idea which the author exploits is the definition of an invariant for a knot \(K\) arising from the dynamics of the homeomorphisms of the punctured disk defined by the braids in \(Bi(K)\). The author introduces an invariant which he denotes by \(\Lambda_K\) and which has the following properties: (i) \(\Lambda_K = 1\) if and only if \(K\) is a torus knot or an iterated torus knot; (i) If \(K\) has braid index \(n\) and is not a torus knot or an iterated torus knot then \(\Lambda_K \geq 2^{1/n}\).
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 57R50
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 590277
zbMATH Keywords: knot \(K\) in the 3-sphere
zbMATH Keywords: homeomorphisms of surfaces
zbMATH Keywords: periodic braid
zbMATH Keywords: hyperbolic knot
zbMATH Keywords: torus knot
zbMATH Keywords: satellite knot
zbMATH Keywords: hyperbolic structure
zbMATH Keywords: braid index set
zbMATH Keywords: pseudo-Anosov
zbMATH Keywords: braided satellite knot
zbMATH Keywords: punctured disk
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
full work available at URL: 0040-9383(94)90014-0
OpenAlex ID: W1994395272
description: scientific article; zbMATH DE number 590277