On derivatives of link polynomials. (English) Zbl 0927.57008

It was proved by J. S. Birman and 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 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.


57M25 Knots and links in the \(3\)-sphere (MSC2010)
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[1] Birman, J.; Lin, X., Knot polynomials and Vassiliev’s invariants, Invent. Math., 111, 225-270 (1993) · Zbl 0812.57011
[2] Brandt, R. D.; Lickorish, W. B.R; Millett, K. C., A polynomial invariant for unoriented knots and links, Invent. Math., 84, 563-573 (1986) · Zbl 0595.57009
[3] Lickorish, W. B.R; Millett, K. C., A polynomial invariant of oriented links, Topology, 26, 107-141 (1987) · Zbl 0608.57009
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