The Kontsevich invariant and the action of the Grothendieck--Teichm\"{u}ller group on $2$-component string links

TL;DR

This paper studies the dependence of the Kontsevich invariant on associators for 2-component string links and demonstrates the non-trivial action of the Grothendieck--Teichmüller group on their algebra, addressing a problem by Furusho.

Contribution

It shows that the Grothendieck--Teichmüller group's unipotent part acts non-trivially on 2-component string links, revealing new insights into the invariant's dependence on associators.

Findings

Kontsevich invariant's dependence on associator for 2-component links

Non-trivial action of Grothendieck--Teichmüller group on string links

Partial resolution of Furusho's problem

Abstract

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--Teichm\"{u}ller group on the algebra of proalgebraic $2$-component string links is non-trivial, which provides a partial answer to a problem posed by Furusho.