Finite type knot invariants are knot invariants satisfying an invariance property with respect to certain local moves. Most knot invariants of interests are of this sort. The collection of finite type invriants form a filtered abelian group. Understanding this object is a fundamental question in knot theory wich was solved by Kontsevich after killing torsion. On the other hand, the algebraic Goodwillie-Weiss tower is a homotopical object whose aim is to compute the homology of the space of knots. It is conjectured that at the level of H_0, this tower produces the universal finite type invariant (in the sense that all other finite type invariants factor uniquely through it). This theorem has been proved modulo torsion by Ismar Volić using the Kontsevich integral. In this talk I will present a new proof of this theorem based on the action of the Grothendieck-Teichmüller group on the little disks operad. This proof has the advantage of also working p-locally in a range. I will finally explain a conjectural generalization of this result to knots in an arbitrary 3 manifolds with boundary.
This is joint work with Pedro Boavida de Brito and Danica Kosanović.