Elementary Combinatorics of the Homflypt Polynomial

We explore F.Jaeger’s state model for the HOMFLYPT polynomial. Reformulated in the language of Gauss diagrams it admits an elementary proof of existence of two HOMFLYPT polynomials for virtual string links which coincide in the case of classical links. In particular, for classical knots it gives a combinatorial proof of existence of the HOMFLYPT polynomial. In the second part of the paper we obtain Gauss diagram formulas for Vassiliev invariants coming from to the HOMFLYPT polynomial. Introduction The HOMFLYPT polynomial P (L) is an invariant of link L. It is defined as the Laurent polynomial in two variables a and z with integer coefficients satisfying the following skein relation and the initial condition: (1) aP ( )− aP ( ) = zP ( ) ; P ( ) = 1 . If L is an unlink with k components then P (L) = ( a−a z k−1 . The existence of such an invariant is a difficult theorem. It was established simultaneously and independently by five groups of authors [HOM, PT]. In this paper we extend the approach of one of them, J.Hoste [H], to virtual links and extract out of it explicit Gauss diagram formulas (see []) for Vassiliev invariants arising from the HOMFLYPT polynomial. ..................................................... 1. HOMFLYPT and descending diagrams The skein relation above allows one to calculate the HOMFLYPT polynomial of a link. Following J.Hoste, this can be done by ordering a link diagram and then transforming it into a descending diagram (see [H] for details). A diagram D is ordered, if its components D1, D2,. . . ,Dk are ordered and on every component a (generic) base point is chosen. An ordered diagram is descending, if Di is above Dj for all i < j and if for every i as we go along Di starting from its base point along the orientation we pass each self-crossing first on the overpass and then on the underpass. An elementary step of the algorithm computing P (L) consists of the following procedure. Suppose that D is an ordered diagram and that the subdiagram D1, . . . ,Di−1 is already descending. We go along Di (starting from the base point) looking for the first crossing which fails to be descending. At such a crossing x we change it using the skein relation. Namely, depending on the sign ε (the local 1 2 SERGEI CHMUTOV AND MICHAEL POLYAK writhe) of the crossing, we express P (D) as (2) P ( ) = aP ( ) + azP ( ) P ( ) = aP ( )− azP ( ) Denote the corresponding diagrams D, D, D. The ordering of D = D induces the ordering of D (in an obvious way); the ordering of D requires some explanation. If x was a crossing of Di with Dj , j > i, then these two components merge into a single component D i of D , with a base point being the base point of Di. If x was a self-crossing of Di, then Di splits into two components: D i , which contains the base point of Di, and D 0 i+1, where we choose the base point in a neighborhood of x. In both cases the order of remaining components shifts accordingly. The diagrams D, D are “more” descending than D. At the next step we apply the same procedure to each of them. Example 1a. For the trefoil 31 the algorithm consists of two steps, illustrated in the figure below. The diagram D appearing in the first step is already descending; the diagram D is not, so the second step is needed to transform it. 1