On the Jones polynomial
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 purely algebraic and combinatorial. (One of the major tasks remaining in understanding these polynomials is to determine the connections, if any, with the standard invariants of algebraic and geometric topology. The only geometric facts used here are that links can be trivialized by changing crossings, and that Reidemeister equivalent projections or Markov equivalent braids determine (ambient) isotopic links.)
scientific article; zbMATH DE number 4009329