A knot-theoretic approach to comparing the Grothendieck-Teichm\"{u}ller and Kashiwara-Vergne groups

TL;DR

This paper explores the deep connections between the Grothendieck-Teichmüller and Kashiwara-Vergne groups using a topological and diagrammatic approach, revealing how these algebraic structures relate through knotted objects and operadic presentations.

Contribution

It provides a topological and diagrammatic analysis of the image of the Grothendieck-Teichmüller groups within Kashiwara-Vergne groups, utilizing operad and tensor category frameworks.

Findings

Establishes a topological perspective on the GRT and KRV groups.

Shows how knotted objects relate to algebraic invariants via operads.

Provides insights into the explicit relationship between associators and KV solutions.

Abstract

Homomorphic expansions are combinatorial invariants of knotted objects, which are universal in the sense that all finite-type (Vassiliev) invariants factor through them. Homomorphic expansions are also important as bridging objects between low-dimensional topology and quantum algebra. For example, homomorphic expansions of parenthesised braids are in one-to-one correspondence with Drinfel'd associators (Bar-Natan 1998), and homomorphic expansions of $w$-foams are in one-to-one correspondence with solutions to the Kashiwara-Vergne (KV) equations (Bar-Natan and the first author, 2017). The sets of Drinfel'd associators and KV solutions are both bi-torsors, with actions by the pro-unipotent Grothendieck-Teichm\"{u}ller and Kashiwara-Vergne groups, respectively. The above correspondences are in fact maps of bi-torsors (Bar-Natan 1998, and the first and third authors with Halacheva 2022). There is a deep relationship between Drinfel'd associators and KV equations--discovered by Alekseev, Enriquez and Torossian in the 2010s--including an explicit formula constructing KV solutions in terms of associators, and an injective map $\rho:\mathsf{GRT}_1 \to \mathsf{KRV}$. This paper is a topological/diagrammatic study of the image of the Grothendieck-Teichm\"{u}ller groups in the Kashiwara-Vergne symmetry groups, using the fact that both parenthesised braids and $w$-foams admit respective finite presentations as an operad and as a tensor category (circuit algebra or prop).