The number of independent Vassiliev invariants in the Homfly and Kauffman polynomials

Abstract

We consider vector spaces ${\mathcal{H}}_{n,\ell }$ and ${\mathcal{F}}_{n,\ell }$ spanned by the degree-$n$ coefficients in power series forms of the Homfly and Kauffman polynomials of links with $\ell$ components. Generalizing previously known formulas, we determine the dimensions of the spaces ${\mathcal{H}}_{n,\ell },{\mathcal{F}}_{n,\ell }$ and ${\mathcal{H}}_{n,\ell }+{\mathcal{F}}_{n,\ell }$ for all values of $n$ and $\ell$. Furthermore, we show that for knots the algebra generated by ${⨁}_n{\mathcal{H}}_{n,1}+{\mathcal{F}}_{n,1}$ is a polynomial algebra with $\dim ({\mathcal{H}}_{n,1}+{\mathcal{F}}_{n,1})-1=n+[n/2]-4$ generators in degree $n\ge 4$ and one generator in degrees 2 and 3.