ABSTRACT
The Kontsevich integral was first invented by M. L. Kontsevich in 1992. Kontsevich used the integration in the way proposed by V. G. Knizhnik and Zamolodchikov. After Kontsevich's original proof, some other sympathetic constructions describing the same knot invariant arose; see the works of P. Cartier and S. Piunikhin. The work by Le and Murakami proposes a concrete method of calculation of the Kontsevich integral. The number of summands is constant for each connected component, but it can vary when passing from one component to another. The part of the knot lying inside the margin between two adjacent critical levels is a set of curves. The chapter explains the Vassiliev module where two knots are taken to be different if they are distinguished by some Vassiliev invariant having order not higher than some fixed order. Besides, each knot can be decomposed into a finite sum of generators of the module.
