Homfly and Kauffman polynomials

We calculate the dimensions of the space of Vassiliev invariants coming from the Homfly polynomial of links, and of the space of Vassiliev invariants coming from the Kauffman polynomial of links. We show that the intersection of these spaces is spanned by the Vassiliev invariants coming from the Jones polynomial and from a polynomial called Υ. We also show that linearly independent Vassiliev invariants of knots of degree ≥ 8 coming from the Homfly or Kauffman polynomial are algebraically independent. Introduction Soon after the discovery of the Jones polynomial V ([Jon]), two 2-parameter generalizations of it were introduced: the Homfly polynomial H ([HOM]) and the Kauffman polynomial F ([Kau]) of oriented links. It was shown by H. Lamaugarny that, in a sense made precise in [Lam], all common specializations of the polynomials H and F can be expressed in terms of V , the number of components of a link, and the linking number. A less known link invariant Υ was studied in [CoG], [Kn1], [Sul] and [Lie]. The link polynomial Υ cannot be obtained from the polynomials H or F of links by a substitution of parameters. Nevertheless, it is natural to regard Υ as a specialization of H and also of F . This can be made precise in the following way, which will turn out to generalize the meaning of a specialization of H or F from [Lam]: Let Vn,l be the vector space of Q-valued Vassiliev invariants of degree n of links with l components. After a suitable substitution of parameters, the polynomial H (resp. F ) can be written as a power series in an indeterminate h, such that the coefficients of h are polynomial-valued Vassiliev invariants pn (resp. qn) of degree n.