A polynomial invariant of knots and links
type: Item
instance of: scholarly article
title: A polynomial invariant of knots and links
zbMATH Open document ID: 0614.57005
author: Jim Hoste
published in: Pacific Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: This paper describes the construction of an invariant of oriented links which is a homogeneous Laurent polynomial of total degree 0 in 3 variables, and which specializes to the Conway and Jones polynomials. Such invariants were discovered independently and almost simultaneously by a number of workers [cf. the joint announcement by \textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239-246 (1985; Zbl 0572.57002)]. Given a projection of the link and a specified resolution to an unlink, a Laurent polynomial may be defined via generalizing Conway identities. It is first shown to be independent of the resolution, and then to be invariant under Reidemeister moves. Thus it is indeed an invariant of the oriented isotopy type of the link. It is also shown that there are distinct prime knots having the same polynomial, and links with homeomorphic complements which have different polynomials.
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 3994561
zbMATH Keywords: oriented links
zbMATH Keywords: Conway and Jones polynomials
zbMATH Keywords: Laurent polynomial
zbMATH Keywords: Conway identities
zbMATH Keywords: Reidemeister moves
zbMATH Keywords: prime knots
zbMATH Keywords: links with homeomorphic complements
reviewed by: Jonathan A. Hillman
MaRDI profile type: Publication
full work available at URL: pjm.1986.124.295
OpenAlex ID: W4237869266
DOI: 10.2140/PJM.1986.124.295
description: scientific article; zbMATH DE number 3994561