Vassiliev invariants of doodles, ornaments, etc.

Alexander B. Merkov

Abstract: In 1990 V. Vassiliev introduced the notion of finite order invariants of knots. These invariants may be thought about as polynomials in the functional space of all invariants. The order of invariants is defined by certain filtration of a resolution of the discriminant set, i.e. of the space of `quasiknots' (smooth non-embeddings of the circle to the $3$-space): the invariants of order $n$ are $0$-cohomologies of the space of knots, dual in some sense to homology of the $n$-th term of the filtration. But after the works of Vassiliev [V90] and Kontsevich [K93] the study of the finite order invariants was reduced to the study of chord diagrams, which represent, in fact, transversal selfintersections of the discriminant, and the homological origins of the theory were nearly forgotten. I'd like to remind the general construction of finite order invariants and the combinatorial objects appearing in the calculation of such invariants. Instead of classification of knots, several variants of classification of plane curves without triple points will be considered. These problems are, in a sense, more generic, because not only transversal selfintersections, but also more complicated singularities of the discriminant, should necessarily be considered. \par On the other hand, diagrams other than the chord diagrams, relevant to classification of knots and plane curves will be constructed, and some recent results by M. Goussarov, M. Polyak, O. Viro, V. Vassiliev and myself will be formulated.

Classification (MSC2000): 57M25; 57M99

Full text of the article:

• Compressed PostScript file (217 kilobytes)
• PDF file (310 kilobytes)