  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).
