BRAIDS, LINK POLYNOMIALS AND A NEW ALGEBRA

Abstract

A class function on the braid group is derived from the Kauffman link invariant. This function is used to construct representations of the braid groups depending on 2 parameters. The decomposition of the corresponding algebras into irreducible components is given and it is shown how they are related to Jones' algebras and to Brauer's centralizer algebras. In [J,3] Vaughan Jones announced the discovery of a new polynomial invariant of knots and links, which bore many similarities to the classical Alexander polynomial, but was seen to detect properties of a link which could not be detected by the Alexander invariants. The discovery was a real surprise, one of those exciting moments in mathematics when two seemingly unrelated disciplines turn out to have deep interconnections. The discovery came about in the following way. Jones' earlier contributions in the area of Operator Algebras had produced, in [J,1], a family of algebras An(t), t E C, indexed by the natural numbers n = 1 , 2, 3, . .. , and equipped with a trace function T: An(t) -C. His algebra An(t) was a quotient of the well-known Hecke algebra of the symmetric group, which we denote by 2 (1, m) to delineate our particular 2-parameter version of it. Jones had discovered, in [J,2], that there were representations of Artin's braid group Bn in the algebra An(t), in fact there were maps Bn 0 Atn(m A(t) from Bn into the multiplicative group of An (t) which factored through Links enter the picture via braids. Each oriented link L in oriented S3 can be represented by a (nonunique) element fl in some braid group Bn. There is an equivalence relation on B = H0= Bn known as Markov equivalence, which determines a 1-1 correspondence between equivalence classes [fl] E B and isotopy types of the associated oriented links Lfl. Jones' discovery was that with a small renormalization his trace function on A o(t) = Hln=l An(t) could be made into a function which lifted to an invariant on Markov classes Received by the editors December 9, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M25; Secondary 20F29, 20C07. The work of the first author was supported in part by NSF grant #DMS-8503758. The work of the second author was supported in part by NSF grant #DMS-8510816. ? 1989 American Mathematical Society 0002-9947/89 $1.00 + $.25 per page