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\title{
  Emergent Version of Drinfeld's Associator Equations
}

\author{Dror~Bar-Natan}

\author{Yusuke Kuno}

\date{First edition Not Yet. This edition \today.}

\thanks{This work was partially supported by NSERC grant RGPIN-2018-04350
and by the Chu Family Foundation (NYC).}


\begin{abstract}
Alekseev, Enriquez, and Torossian \cite{AlekseevEnriquezTorossian:ExplicitSolutions}, continuing Alekseev and Torossian \cite{AlekseevTorossian:KashiwaraVergne}, show how from a solution of the Drinfel'd pentagon equation one may construct a solution of the Kashiwara-Vergne (KV) equations \cite{KashiwaraVergne:Conjecture}. In this paper we show that solutions of a weak version of the pentagon equations that we call {\em the emergent pentagon equations} are sufficient for the same task, the construction of solutions of the KV equations.

The emergent pentagon equations arise within a natural topological context of {\em emergent tangles}, which we discuss. Our results are adjacent to the results of \cite{LesDiablerets-2208, BDHLS} on the relationship between emergent tangles and the Goldman-Turaev Lie bialgebra. We hope that in time our results will play a role in relating several bodies of work, on Drinfel'd associators, Kashiwara-Vergne equations, and on expansions for classical tangles, for w-tangles, and for the Goldman-Turaev Lie bialgebra.
\end{abstract}

\maketitle

\section{Introduction}

\begin{thebibliography}{BDHLS}

\bibitem[AT]{AlekseevTorossian:KashiwaraVergne} A.~Alekseev and
  C.~Torossian,
  {\em The Kashiwara-Vergne conjecture and Drinfel'd's associators,}
   Annals of Mathematics, {\bf 175} (2012) 415--463, \arXiv{0802.4300}.

\bibitem[AET]{AlekseevEnriquezTorossian:ExplicitSolutions} A.~Alekseev, 
B.~Enriquez, and C.~Torossian,
   {\em Drinfel'd's associators, braid groups and an explicit solution of 
the Kashiwara-Vergne equations,}
    Publications Math\'ematiques de L'IH\'ES, {\bf 112-1} (2010) 143--189, 
\arXiv{0903.4067}.

\bibitem[BN1]{LesDiablerets-2208} Dror Bar-Natan,
  {\em Tangles in a Pole Dance Studio: A Reading of Massuyeau, Alekseev, and Naef,}
  talk given in {\em Algebra, Topology and the Grothendieck-Teichmüller group}, Les Diablerets, September 2022. Handout and video at \url{http://drorbn.net/ld22}.

\bibitem[BDHLS]{BDHLS} Dror~Bar-Natan, Zsuzsanna Dancso, Tamara Hogan, Jessica Liu, and Nancy Scherich,
  {\em Goldman-Turaev formality from the Kontsevitch integral,}
  in preparation.

\bibitem[KV]{KashiwaraVergne:Conjecture} M.~Kashiwara and M.~Vergne,
  \href{http://www.springerlink.com/content/v73014gx14084624/}{{\em The
    Campbell-Hausdorff Formula and Invariant Hyperfunctions,}}
  Invent.{} Math.{} {\bf 47} (1978) 249--272.

\end{thebibliography}

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