On Fox's congruence classes of knots. II
type: Item
instance of: scholarly article
title: On Fox's congruence classes of knots. II
zbMATH Open document ID: 0705.57006
author: Yasutaka Nakanishi
published in: Osaka Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: [For Part I by the author and \textit{S. Suzuki} see ibid. 24, 217-225 (1985; Zbl 0654.57003).] In ``Congruence classes of knots'' [Osaka Math. J. 10, 37-41 (1958; Zbl 0084.192)], \textit{R. H. Fox} defined (n,q) congruence of knots by means of what is now called P/Q surgery, where \(P=1\) and certain conditions on the surgery are satisfied. Congruence generalizes to links and generates an equivalence relation among link types. Some theorems the author proves are: Theorem 2. Let \(n>0\), \(q>0\), (n,q)\(\neq (2,1),(2,2)\). For every such (n,q) and every \(\mu\geq 1\) there exist infinitely many distinct classes of \(\mu\)-component links which are inequivalent mod(n,q). Theorem 3. Two links with the same number of components are congruent mod(2,1) if and only if the components may be ordered so that each pair of corresponding components of each link have the same linking numbers mod 2. Corollary. All knot types are congruent mod(2,1). The author shows the Borromean rings and a 3-component trivial link are not congruent mod(2,2).
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 4157493
zbMATH Keywords: P/Q surgery
zbMATH Keywords: links
zbMATH Keywords: \(\mu \) -component links
zbMATH Keywords: linking numbers
zbMATH Keywords: Borromean rings
reviewed by: Lee P. Neuwirth
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4157493