%From: "Victor A. Vassiliev"%Date: Thu, 20 Nov 97 20:39:36 +0300 \documentstyle[12pt]{article} \begin{document} The order $k$ knot invariants usually are characterized by the values of their indices at elementary singular knots of complexity $k$, i.e. at immersed circles with $k$ transverse self-intersections. Similar generalized indices can be defined for arbitrary maps of the circle into ${\bf R}^3$. They are closely related with the combinatorial graph theory: e.g., for a curve with one point of $m$-fold selfintersection this index takes values in the homology group of the complex of two-connected graphs with $m$ vertices. More generally, for any cohomology class of the space of knots in ${\bf R}^n$, $n \ge 3$ (of finite order if $n=3$) we define its {\em principal symbol}, lying in the cohomology group of a certain finite-dimensional configuration space and characterizing this class up to classes of smaller filtrations. These notions allow us to simplify the calculation of cohomology groups of spaces of knots in ${\bf R}^3$. Using them we calculate all cohomology groups of order $\le 3$ (resp., $\le 2$) of spaces of non-compact (compact) knots in ${\bf R}^n$ for arbitrary $n$. Recently it was proved (A.~Bj\"orner with co-authors and, independently, V.~Turchin) that the crucial here homology group of complexes of two-connected graphs with $m$ vertices is trivial in all dimensions other than $2m-4$ and is isomorphic to ${\bf Z}^{(m-2)!}$ in this dimensions. Also V.Turchin has established the natural isomorphism between the calculi of two-connected graphs and Chinese chord diagrams with exactly $m$ legs. \end{document}