On the Jones polynomial
type: Item
instance of: scholarly article
title: On the Jones polynomial
zbMATH Open document ID: 0622.57002
author: Pierre de la Harpe
author: Claude Weber
author: Michel Kervaire
published in: L'Enseignement Mathématique. 2e Série
publication date: 1986-01-01T00:00:00Z
review text: The ''homfly'' polynomial of an oriented link is a 2-variable Laurent polynomial which simultaneously generalizes the Alexander-Conway and Jones polynomials. It was discovered independently and almost simultaneously by four groups; the name ''homfly'' is formed from the initial letters of the surnames of those involved [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New. Ser. 12, 239-246 (1985; Zbl 0572.57002)]. This paper presents a lucid exposition of this invariant from the point of view of Ocneanu, who extended Jones original approach to his 1- variable polynomial. It if first shown that if such a 2-variable polynomial exists, it is uniquely determined by having value 1 for the unknot, and satisfying ''skein identities'', like those discovered for the Alexander polynomial by Conway. Moreover it is universal among invariants of oriented links taking values in a commutative ring and satisfying such identities. Hecke algebras and trace functions are then used to construct a braid invariant which is shown to be invariant under Markov moves, and so is a link-invariant. After a change of variables this becomes a Laurent polynomial with the required properties. After giving some basic properties, such as behaviour under changes of orientation and forming sums of links, and relation to the single- variable polynomials, Kauffman's approach to the Jones polynomial in terms of combinatorial properties of link projections invariant under Reidemeister moves is described. This is followed by a discussion of some conjectures on minimal crossing number, alternating projections, etc., surviving from the prehistory of knot theory, which have at last been largely settled by Kauffman and Murasugi, using the Jones polynomial. In the final section the original derivation of the Jones polynomial from work on Von Neumann algebras is outlined. While this is certainly of interest, no knowledge of Von Neumann algebras is needed to understand the main body of the paper. The arguments are …
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 46L10
zbMATH DE Number: 4009329
zbMATH Keywords: homfly polynomial of an oriented link
zbMATH Keywords: Jones polynomials
zbMATH Keywords: skein identities
zbMATH Keywords: Alexander polynomial
zbMATH Keywords: Hecke algebras
zbMATH Keywords: trace functions
zbMATH Keywords: braid invariant
zbMATH Keywords: Markov moves
zbMATH Keywords: link-invariant
zbMATH Keywords: Reidemeister moves
zbMATH Keywords: minimal crossing number
zbMATH Keywords: alternating projections
zbMATH Keywords: Von Neumann algebras
reviewed by: Jonathan A. Hillman
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4009329