The universal finite-type invariant for braids, with integer coefficients (Q5960435)

The universal Vassiliev invariant of knots and links is a finite-type invariant with rational coefficients that was constructed by Kontsevich in order to unify the theory of invariants of knots and links in the three-sphere. The paper under review gives a simple and explicit construction of a universal finite type invariant with integer coefficients for braids. More precisely, it constructs a filtered map \(M\) from the group ring of the Artin braid group with \(n\) strings to a certain graded algebra inducing an isomorphism at the graded level. Although the idea of the existence of such an invariant follows naturally from the work of Kontsevich, the techniques used here are different. The proof of the main result of this paper is done in two main steps. The first step is based on a result of Quillen which establishes a strong connection between the \(I\)-adic filtration of an arbitrary group ring and the lower central series filtration of the group. The second step deals with the \(I\)-adic filtration of the group ring of the pure braid group.