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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.

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010)
##### Keywords:
skein relation; Vassiliev’s invariants; Homfly polynomial
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