Epimorphisms of knot groups onto free products
type: Item
instance of: scholarly article
title: Epimorphisms of knot groups onto free products
zbMATH Open document ID: 1025.57010
DOI: 10.1016/S0040-9383(02)00087-3
author: F. González-Acuña
author: Arturo Ramírez
published in: Topology
publication date: 2003-07-29T00:00:00Z
review text: Imposing conditions on a closed surface in \(S^3\) containing a knot \(k\) leads to the definition of properties Q, Q*, and Q**. Roughly speaking, property Q is a crude generalization of torus knots. The following theorem is proved and is representative of other results in the paper. The knot \(k\) has property Q iff any one of the following is true: (1) There exists an epimorphism from the group of the knot to \(\mathbb{Z}_a*\mathbb{Z}_b\) sending a meridian to an element of length 2. (2) There exists an \(H_1\) splitting surface for \(k\) whose boundary has slope with absolute value 2. (3) There exists an incompressible \(H_1\) splitting surface for \(k\) whose boundary has slope with absolute value 2. (4) There exists an epimorphism from the group of the knot to the group of a torus knot sending a meridian to a meridian. A corollary to this theorem is that if a knot has property Q then its Alexander Polynomial is divisible by the Alexander Polynomial of a torus knot. The paper contains a number of examples.
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M07
zbMATH DE Number: 1956002
zbMATH Keywords: knot groups
zbMATH Keywords: interpolating surface
zbMATH Keywords: splitting surface
zbMATH Keywords: free product
zbMATH Keywords: property Q
reviewed by: Lee P. Neuwirth
MaRDI profile type: Publication
full work available at URL: s0040-9383(02)00087-3
OpenAlex ID: W2068211956
description: scientific article; zbMATH DE number 1956002