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Link polynomials of higher order. (English) Zbl 0903.57002
Polynomials \(P_L (x, y, z)\) of links (singular or non-singular) that are related to the Homfly polynomial and Vassiliev’s invariants are studied. The invariants satisfy the skein relations \(P_{L_{\times \times}} = 0\), and \[ P_{L_{\times}} = x P_{L_{+}} + y P_{L_{-}} + z P_{L_{0}} , \] where \(L_{\times}\), \(L_{+}\), \(L_{-}\), and \(L_{0}\) denote links that are the same, except inside a 3-ball where they are related by standard crossing changes.
Let \({\mathcal L}^n\) be the set of oriented links with \(n\) crossing vertices. The author gives descriptions of zeroth order skein invariants on \({\mathcal L}^1\) and first order invariants on \({\mathcal L}^0\). In particular, in the case \(x=1\), \(y=-1\), and \(z=0\) the invariant \(P_K\) yields a Vassiliev invariant of order 1 of a knot \(K\). Some properties of the higher order link polynomials and possible applications are discussed.

57M25 Knots and links in the \(3\)-sphere (MSC2010)
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