Linear skein theory and link polynomials
type: Item
instance of: scholarly article
title: Linear skein theory and link polynomials
zbMATH Open document ID: 0638.57005
DOI: 10.1016/0166-8641(87)90091-5
published in: Topology and its Applications
publication date: 1987-01-01T00:00:00Z
review text: It is well known that the Jones polynomial of an oriented link is fairly stable under the reversal of the orientation of a component. In fact, let \(V_{L_ 1}(t)\) and \(V_{L_ 2}(t)\) be the Jones polynomials of two oriented links \(L_ 1\) and \(L_ 2\), one of which, say \(L_ 2\), is obtained from \(L_ 1\) by reversing the orientation of some of its components. Then \(V_{L_ 1}(t)=t^{\lambda}V_{L_ 1}(t)\) from some integer. In this paper, this result is generalized to the generalized Jones polynomials from a slightly different point of view. Let \(P_ L(1,m)\) denote the polynomial defined by Lickorish and Millett, and \(F_ L(a,z)\) the polynomial defined by L. Kauffman. Let L be an oriented link of n components \(K_ 1,...,K_ n\). Choose one component, say \(K_ n\) and construct an oriented satellite link \(K^*_ n\) (with \(K_ n\) as a companion) in a small tubular neighborhood of \(K_ n\). Thus we obtain a new oriented link \(L_ 1=K_ 1\cup...\cup K_{n-1}\cup K^*_ n\). Now by reversing the orientations of all components of \(K^*_ n\), we will obtain another oriented link \(L_ 2\). Then it is proved that \(P_{L_ 1}(l,m)=P_{L_ 2}(l,m)\) and \(F_{L_ 1}(a,z)=F_{L_ 2}(a,z)\) (Theorems 2 and 3). The author also shows that if L is a mutant of L', then \(F_ L(a,z)=F_{L'}(a,z)\). The paper includes the matrix formula to evaluate \(F_ L(a,z)\) for a 2-bridge link L.
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 4040851
zbMATH Keywords: satellite knot
zbMATH Keywords: mutant
zbMATH Keywords: polynomial of an oriented link
reviewed by: K. Murasugi
MaRDI profile type: Publication
full work available at URL: 0166-8641(87)90091-5
OpenAlex ID: W2146931846
cites work: Q3687355
cites work: A polynomial invariant for unoriented knots and links
cites work: A new polynomial invariant of knots and links
cites work: Examples on polynomial invariants of knots and links
cites work: An Invariant of Regular Isotopy
cites work: A relationship between link polynomials
cites work: Polynomials of 2-Cable-Like Links
cites work: A polynomial invariant of oriented links
cites work: Some evaluations of link polynomials
cites work: The 2-variable polynomial of cable knots
author: W. B. Raymond Lickorish
description: scientific article; zbMATH DE number 4040851