The Kontsevich invariant and the action of the Grothendieck--Teichmüller group on $2$-component string links

The Kontsevich invariant of links is independent of the choice of associator, whereas for tangles this is not the case in general. In this paper, we focus on $2$-component string links and investigate to what extent the Kontsevich invariant depends on the choice of associator. As an application, we show that the action of the unipotent part of the Grothendieck–Teichmüller group on the algebra of proalgebraic $2$-component string links is non-trivial, which provides a partial answer to a problem posed by Furusho.