The Number of Independent Vassiliev Invariants in the Homfly and Kauffman Polynomials

We consider vector spaces $\Hvnl$ and $\Fvnl$ 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 $\Hvnl$, $\Fvnl$ and $\Hvnl+\Fvnl$ for all values of $n$ and $\ell$. Furthermore, we show that for knots the algebra generated by $\bigoplus_n \Hvne+\Fvne$ is a polynomial algebra with $\dim(\Hvne+\Fvne)-1=n+[n/2]-4$ generators in degree $n\geq 4$ and one generator in degrees $2$ and $3$.

1991 Mathematics Subject Classification: 57M25.

Keywords and Phrases: Vassiliev invariants, link polynomials, Brauer algebra, Vogelīs algebra, dimensions.

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