Knotting trivial knots and resulting knot types
type: Item
instance of: scholarly article
title: Knotting trivial knots and resulting knot types
zbMATH Open document ID: 0788.57004
author: Kimihiko Motegi
published in: Pacific Journal of Mathematics
publication date: 1994-06-05T00:00:00Z
review text: Let \((V,K)\) be a pattern (i.e. \(V\) is a standardly embedded solid torus in oriented \(S^ 3\) and \(K\) a knot in \(V\)) and \(f\) an orientation preserving embedding from \(V\) into \(S^ 3\) such that \(f(V)\) is knotted. Answers to the following questions are given depending upon whether the winding number of \(K_ 2\) in \(V\) is zero or not. (1) Suppose that \(K_ 1\) is unknotted and \(K_ 2\) is knotted in \(S^ 3\). Can \(f(K_ 1)\) be ambient isotopic to \(f(K_ 2)\) in \(S^ 3\) for some embedding \(f: V\hookrightarrow S^ 3\)? (2) Suppose that \(K_ 1\) and \(K_ 2\) are both unknotted in \(S^ 3\). How are \((V,K_ 1)\) and \((V,K_ 2)\) related if \(f(K_ 1)\) is ambient isotopic to \(f(K_ 2)\) in \(S^ 3\) for some embedding \(f: V\hookrightarrow S^ 3\)?
reviewed by: Kimihiko Motegi
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 540671
zbMATH Keywords: satellite knot
zbMATH Keywords: wrapping number
zbMATH Keywords: pattern
zbMATH Keywords: solid torus
zbMATH Keywords: winding number
zbMATH Keywords: unknotted
MaRDI profile type: Publication
full work available at URL: pjm.1993.161.371
OpenAlex ID: W2068014717
DOI: 10.2140/PJM.1993.161.371
description: scientific article; zbMATH DE number 540671