On derivatives of link polynomials
It was proved by \textit{J. S. Birman} and \textit{X.-S. Lin} [Invent. Math. 111, No. 2, 225-270 (1993; Zbl 0812.57011)] that the link polynomials all yield Vassiliev invariants by making a suitable change of variables and then taking coefficients of the Taylor expansion. These coefficients are essentially higher derivatives evaluated at a special value. The authors consider the higher order link polynomials \(P_L(x,y,z)\), introduced by \textit{Y. Rong} [J. Lond. Math. Soc., II. Ser. 56, No. 1, 189-208 (1997; Zbl 0903.57002)] and study their partial derivatives with respect to \(x,y\), and \(z\). They prove that each partial derivative of an \(n\)-th order Homfly polynomial is an \((n+1)\)-th order Homfly polynomial. In particular, the partial derivatives of the Homfly polynomial yield first order Homfly polynomials. Similar constructions for other link polynomials complete the paper.
scientific article; zbMATH DE number 1307963