%From: "Victor A. Vassiliev"
%Date: Thu, 20 Nov 97 20:39:36 +0300

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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.

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