Vassiliev spaces and classical invariants
In this nice and easily readible article, the author shows how a number of `classical' invariants from various branches of mathematics can be introduced analogously to Vassiliev's knot invariants -- that is by choosing an appropriate discriminant space and identifying elements of it's ``dual'', the so-called ``Vassiliev space''. The author distinguishes the cases of finite- and infinite-dimensional discriminants and those of discriminants with only canonical and with non-canonical singular points. In the canonical case, Vassiliev invariants always exist and are determined by their values on the discriminant's complement (Differentiation lemma). Examples are given to illustrate the various cases, namely the degree of a point with respect to a plane curve, the index of a quadratic form, the Whitney index, the Gauss linking number, Arnold's invariants of plane curves and the Maslov index and the Bennequin number of a Legendrian knot. Other examples are hinted at, as well as an example of an invariant which does not fit into this scheme. For the reader particularly interested in knot theory, it should be remarked that the article does not explain the technical difficulties arising in constructing the (original) Vassiliev knot invariants, but that it gives something like a ``good first impression'' of the method.
scientific article; zbMATH DE number 1208182