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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.

MSC:
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
[4] Y. Rong, Higher order link polynomials, J. Lond. Math. Soc., to appear. · Zbl 0903.57002
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