ABSTRACT
In 1970 Conway proposed an inductive way to construct Alexander's polynomials for knots and links which is simpler from a computation viewpoint. He observed that for three oriented links whose diagrams differ only in a neighborhood of a point and such that inside that neighborhood the links have views as in Figure 36.1, the polynomials satisfy a simple relation. Denote by L+, LP, L. the links corresponding to the first, second and third pictures of Figure 36.1. Let el, t.2 be a positively oriented basis on the diagram plane. For the link L+ the vector el goes through the point of selfintersection above e2 and for the link L_ the vector el goes below e2. Then the Alexander polynomial, when suitably normalized, satisfies the following equation:
In 1983 Jones constructed a new polynomial VL( t ) using the braid theory and the representation of braid groups in the Hecke algebras [71]. It was a great surprise to topologists. This polynomial opened the way to solving some old problems in knot theory. For example, this new polynomial distinguishes, for some cases, the knot and its mirror image. This polynomial satisfies the condition
In 1985, almost at the same time, four groups of mathematicians, encouraged by Jones' success, obtained one and the same result. They proved the existence and studied the properties of a new polynomial invariant for knots and links. Their results were gathered by the editors of Bull. Amer. Math. Soc. into a joint paper by six authors, P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorich, K. Millett and A. Ocneanu, and published [74]. The paper defined a homogeneous polynomial of three variables PL(x, y , Z ) with positive and negative degrees of these variables X, y, z. The polynomial was called the HOMFLY polynomial.
