\documentclass[12pt]{amsart}
\usepackage{amsmath,amssymb,mathtools}
\usepackage{tikz,tikz-cd}
\usetikzlibrary{decorations.markings,arrows.meta}

\begin{document}

\title[Diagrammatic objects and the Turaev cobracket]{Diagrammatic Objects Pine for Specializations. And the Turaev Cobracket?}
\author{}
\maketitle

Let $\Sigma$ be a compact oriented surface with first homology
$H = H_1(\Sigma;\mathbb Q)$.
The $\mathbb Q$-vector space spanned by free homotopy classes of loops on~$\Sigma$
carries the Goldman bracket $\{\ ,\ \}$ and the Turaev cobracket $\delta$,
forming the Goldman--Turaev Lie bialgebra.
After $I$-adic completion and passage to the associated graded with respect to the
augmentation ideal of $\mathbb Q\pi_1(\Sigma)$, one obtains a graded Lie bialgebra
on
\[
\mathrm{Cyc}(H) := T(H)/[T(H),T(H)],
\]
where the grading is by word length.

\vskip.4em
\noindent{\bf 1. The graded cobracket.}
Write $\mathrm{Cyc}^m(H)$ for cyclic words of length $m$.
The leading term
\[
\delta_{-1} : \mathrm{Cyc}^m(H) \longrightarrow
\bigoplus_{p+q = m-1} \mathrm{Cyc}^p(H)\otimes \mathrm{Cyc}^q(H)
\]
is of degree $-1$: it lowers total length by one.
For genus $0$, $\delta_{-1}$ is the
necklace cobracket of Schedler:
it is given by summing over ordered pairs of positions in a cyclic word,
inserting the intersection form and cutting the word into two cyclic pieces.
(Any explicit formula we write should reflect this degree $-1$ behavior.)

\begin{center}
\begin{tikzpicture}[scale=1.0,baseline={(current bounding box.center)}]
  % main circle
  \draw[thick] (0,0) circle (1);
  % beads
  \foreach \ang in {15,75,135,195,255,315} {
    \filldraw[fill=white] (\ang:1) circle (2pt);
  }
  % cut marks
  \draw[very thick,red] (30:1.05) -- (30:1.35);
  \draw[very thick,red] (210:1.05) -- (210:1.35);
  % omega
  \node[red] at (1.3,0.1) {$\omega$};
\end{tikzpicture}

{\small The graded cobracket cuts a beaded circle and contracts one pair of beads, lowering degree by 1.}
\end{center}

\vskip.4em
\noindent{\bf 2. What is known: noncommutative avatars.}
For suitable group-like (``special'') expansions, the Goldman bracket and Turaev
cobracket admit algebraic descriptions in terms of
Van den Bergh-style double brackets and a noncommutative divergence cocycle.
In genus $0$, Alekseev--Kawazumi--Kuno--Naef show that
$\delta$ can be reconstructed from such data, and that choosing expansions
compatible with the framed Turaev cobracket is essentially equivalent to solving
the Kashiwara--Vergne problem.
Relatedly, Kawazumi and Hain interpret parts of regular-homotopy and framed
versions of $\delta$ via divergence-type cocycles.

\vskip.4em
\noindent{\bf 3. What is missing.}
These descriptions are powerful but not yet a single, canonical
``geometric divergence operator on a moduli space'' that one could point to and
say, without qualifiers,
\[
\text{``the Turaev cobracket \emph{is} this specialization''}.
\]
The graded Turaev cobracket sits as a natural diagrammatic operation of degree $-1$,
beautifully constrained and algebraically understood in several frameworks,
but its interpretation as a straightforward geometric or Lie-theoretic
operation remains, at best, indirect and expansion-dependent.

\vskip.4em
\noindent{\bf 4. Tagline.}
\emph{Diagrammatic objects pine for specializations.
For the Turaev cobracket, the wait is not convincingly over.}

\end{document}
