Segment-arrow diagrams and invariants of ornaments

A. B. Merkov

Sbornik. Mathematics, Tome 191 (2000) no. 11, pp. 1635-1666

Résumé

An ornament is a finite collection of closed oriented curves in the plane no three of which have common points. Homotopy invariants of ornaments are considered. Similarly to the case of knot classification, all invariants of ornaments are equal to the linking numbers with appropriate cycles in the discriminant, that is, in the set of collections of curves with forbidden intersections. Finite-order (or Vassiliev) invariants are those for which the corresponding cycle can be described in terms of finitely many strata in the natural stratification of the discriminant by the types of forbidden points. The calculation of these invariants is reduced to the calculation of certain cohomological spectral sequences. A new explicit combinatorial construction of a series of finite order invariants of ornaments is presented. It is shown that some previously known series of finite order invariants are contained in this series, which can also be expressed in terms of cohomology classes of natural finite-dimensional topological spaces.

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