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

TL;DR

The paper analyzes how the Kontsevich invariant for 2-component string links depends on the associator, proving independence up to degree 6 and demonstrating a controlled, explicit dependence at higher degrees via degree-8 phenomena. It connects this dependence to twists by degree-3 diagrams and to the action of the unipotent part of the Grothendieck--Teichmüller group on proalgebraic string links, yielding a nontrivial GT_1 action in the 2-component case. A central algebra A(ownarrowownarrow) is studied to establish noncommutativity results essential for the degree-8 construction, and the paper further shows that a broad family of weight systems yields invariants independent of associator choice when composed with Z_Φ. Together, these results illuminate the interplay between associators, GT actions, and finite-type (Vassiliev) invariants in the realm of 2-component string links, with implications for motivic Galois actions on proalgebraic tangles.

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--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.

Figures (5)

• Figure 1: An example of a $2$-component string link.
• Figure 2: The $\mathit{GT}(\mathbb{K})$-action on elementary tangles. Here, $\sigma=(\lambda, f) \in \mathit{GT}(\mathbb{K})$ and we set $f_{1\cdots i-1, i, i+1} \coloneqq f(x_{1 \cdots i-1, i}, x_{i i+1}) = f(x_{1i} x_{2i} \cdots x_{i-1i}, x_{ii+1})$ when $i \geq 2$ and $f_{1\cdots i-1, i, i+1}=1$ when $i=1$. Note that $\exp(\frac{\lambda-1}{2} \log (x_{i, i+1}))$ is well-defined in $\widehat{\mathbb{K}[\mathit{\mathit{PB}}_n]}$. The element $\nu_f \in \widehat{\mathbb{K}[\mathcal{SL}_1]}$ is defined in the same way as $\nu$ in Section \ref{['subsec:Kontsevich']}, by replacing $\Phi$ with $f$.
• Figure 3: The $\mathit{GRT}(\mathbb{K})$-action on fundamental infinitesimal tangles. Here, $\sigma=(c,g) \in \mathit{GRT}(\mathbb{K})$ and we set $g_{1\cdots i-1, i, i+1} = \Delta_1^{i-2} g$ when $i \geq 2$ and $g_{1\cdots i-1, i, i+1}=1$ when $i=1$. The element $\nu_g \in \widehat{\mathcal{A}}(\downarrow)_{\mathbb{K}}$ is defined in the same way as $\nu$ in Section \ref{['subsec:Kontsevich']}, by replacing $\Phi$ with $g$.
• Figure 4: Three possible non-closed curves in the resolution of a Jacobi diagram in $\mathcal{A}(\downarrow \downarrow)$.
• Figure 5: Typical examples of the cases (a), (b), and (c).

Theorems & Definitions (51)

• Proposition 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Proposition 1.6
• Remark 2.1
• Example 2.2
• Remark 2.3
• Example 2.4