\documentclass[11pt]{amsart}


\title[Comments]{Comments on the paper ``Goldman--Turaev formality from the Kontsevich integral''}


\newcommand{\change}[4]{\noindent P.#1, l.#2: ``#3'' $\leadsto$ ``#4''\\}
\newcommand{\com}[2]{\noindent  #1: #2\\}
\newcommand{\compl}[3]{\noindent  P.#1, l.#2: #3\\}


\begin{document}

\maketitle

\vspace{-0.5cm}

\section{Comments and questions}

\com{Eq. (2.1)}{This is a rather sophisticated way to use the ``Snake Lemma'' for the following:
Given a homomorphism $\lambda: B \to E$ and two submodules $A\leq B$, $D\leq E$,
if $\lambda(A)=0$ and $\lambda(B) \subset D$, then $\lambda$ induces a map $\eta:B/A\to D, \{b\} \mapsto \lambda(b)$.}

\com{Eq. (2.1)}{It is written two lines before that ``$\lambda$ induces a unique map $\eta$'': since the unicity
is used several times in the sequel, it would be  better to specify in which sense $\eta$ is induced by $\lambda$
(i.e., have the  commutative quadrilateral with vertices $B,C,D,E$). 
Thus, the reader will understand what you mean, for instance in the proof of Cor. 2.4, when you write ``makes the diagram commute.''}

\com{Example 2.1}{As it is defined here, $A/M_1$ is not the abelianization of $A$, but it is zero. Indexes have to be shifted in some way.}

\com{Lemma 3.1}{I think that the order of composition has been reversed: what we wish is a filtered map $Z$
such that $\hbox{Gr} Z \circ \psi$ is the identity of $\mathcal{C}$; since $\psi$ is surjective, this identity implies
that $\psi$ is an isomorphism, and so  is $\hbox{Gr} Z$. Furthermore, in practice, this is what happens: we have a presumed
``universal'' invariant $Z$, we propose a ``surgery'' map $\psi$ and we compute $\hbox{Gr} Z \circ \psi$.
(Same remark in Lemma 4.8.)}


\com{P. 11}{What you call ``enhanced cobracket'' here is actually a variation of the self-intersection operation $\mu$
that Turaev introduced long before his cobracket (\emph{Math. Sb.} 1978): 
see \S 3.2 of [Mas18] and the operation $\delta_{\vec{\mu}}$ that is defined there. I think that those two references should be added.}

\com{\S 3.3}{Some  references could be added here too: first, the Lie bialgebra structure that is described here on $\vert \hbox{FA}\vert$
is known as the ``necklace Lie bialgebra'', whose cobracket has been introduced by Schedler (\emph{IMRN} 2005);
second, that the associated graded of the Goldman--Turaev Lie bialgebra of a punctured disk is the necklace Lie bialgebra
was also observed in [Mas18, \S5.4 \& \S 7].}

\com{Pages 32--33}{It took me time to understand your description of the maps $\lambda_a$ and $\lambda_d$.
I would have said things as follows: (1) For $\gamma \in \pi$,  define 
$\lambda_a( \gamma)\in \tilde{\mathcal{T}}(\cap)$ as the ascending lift of $\gamma$ 
(which is a well-defined unframed bottom knot), and then set its framing number to be $+1$; 
the map $\lambda_d$ is defined similarly; (2) Explain how to draw a ``floor diagram'' of 
$\lambda_a( \gamma)$  (resp$.$ $\lambda_d( \gamma)$) 
using the regular homotopy class $\tilde{\gamma}$ in $D_p$ 
and inserting the appropriate ``kinks''  to get the framing number equal to $+1$ 
(here, Whitney's formula is needed to relate the rotation number to the algebraic self-intersection 
of a regular curve in the disk); (3) compare the  ``floor diagrams'' of $\lambda_a( \gamma)$ 
and $\lambda_d( \gamma)$ that are obtained in this way.}

\com{Page 37}{The lack of formality of the self-intersection map $\mu$ (or its variant $\vec{\mu}$)
is also proved in [Mas18, Th. 7.1]: the correcting term (which is denoted there by $q$)
is expressed in terms of the Gamma function of the associator underlying 
the construction of the Kontsevich integral.}

\com{Page 38, footnote}{It could be specified, somewhere in \S3 or \S4, which version of the Kontsevich integral is used in the paper.
In particular, it could be said why you prefer to work with the ``analytic'' construction rather than the ``combinatorial'' construction.}

\com{Page 44}{I don't understand the ``+'' sign in cl$(\vert 1\vert \otimes B + \vert B \vert \otimes1)$ at the very end of the proof.}

\section{Typos and other minor comments}

\change{4}{6}{Another approach}{Another approach to}

\change{4}{-19}{biaglebra}{bialgebra}

\change{4}{-17}{Gwenel}{Gwena\"el}

\change{5}{-3}{$\pi(B^n)$}{$\pi(B^i)$}

\compl{5}{6}{Repetition in the sentence ``Therefore the Kontsevich integral therefore (...)''.}

\change{7}{-12}{Kontsevich lntegral}{Kontsevich integral}

\change{7}{-8}{presentated}{presented}

\change{9}{-13}{which is contracts}{which contracts}

\change{10}{8}{Let $D_p$ denote}{Let $D_p$ denote the}

\compl{10}{-10}{It should be specified that the path $\nu$ is from $\star$ to $\bullet$ in the negative direction of
the oriented boundary of $D_p$. Moreover, the second base point has three different notations in your paper:
$\star$, $\ast$ and this kind of ``virus'' symbol in your pictures.}

\change{12}{13}{There I-adic filtration}{The I-adic filtration}

\compl{13}{8}{The text is here refering to Figure 7, but the notations ``$z$ and $w$'' are not shown in that figure.}

\change{13}{-5}{$As_p$}{FA}

\change{14}{10}{$\vert As_p\vert$}{$\vert \hbox{FA}\vert$}

\change{15}{6}{schematic diagram}{a schematic diagram}

\change{18}{-9}{that it $\psi$}{that $\psi$}

\change{19}{-17}{by the same names}{denoted by the same names}

\change{19}{-3}{as in see Figure}{as in Figure}

\compl{19}{-1}{There is an unexpected ``18'' here.}

\change{20}{8}{subscripts}{superscripts}

\compl{21}{-13}{Two occurrences of $\psi$ should be changed to $\psi_\nabla$.}

\compl{22}{8}{This third identity here should be decorated by a ``$\nabla$'', since it is only true in the Conway quotient.}

\change{22}{-9}{bialebra}{bialgebra}

\change{22}{-7}{$\tilde{\mathcal{T}}/\tilde{\mathcal{T}}^n$}{$\mathbb{C}\tilde{\mathcal{T}}/\tilde{\mathcal{T}}^n$}

\change{24}{-8}{Goldman Bracket}{Goldman bracket}

\compl{24}{-1}{It might be specified that the domain of $\beta$ is endowed with the t-filtration.}

\change{25}{-19}{pole-stand}{pole-strand}

\compl{25}{-6}{The subspace $\tilde{\mathcal{A}}^{\geq i}(O)$ could be denoted $\tilde{\mathcal{A}}^{ i}(O)$ 
for coherence with your other notations for filtrations.}

\change{26}{l.1}{$x_3^2x^2x_1^2x_2x_3$}{$x_3^2x_2x_1^2x_2x_3$}

\compl{26}{-12}{It seems that a circle skeleton component is missing in the target of the map $\lambda_1$.}

\change{27}{5}{be a crossing}{is a crossing}

\change{27}{6}{speparate components}{separate components}

\change{27}{-9}{identifies}{and, so, identifies}

\change{27}{-8}{lies in in}{lies in}

\change{28}{9}{Golman bracket}{Goldman bracket}

\compl{31}{-12}{I would say that there is no Conway quotient on the source of this map $\beta$.}

\compl{31}{-2}{The skeleton does not seem to be correct in this description of the kernel.}

\compl{32}{5}{The Conway quotient does not seem to be relevant in considering $\tilde{\mathcal{T}}^{/1}$. (Here and twice below
in the proof of Prop. 5.11.)}

\compl{34}{7}{$k:=1$ and $\ell:=1$ in this occurrence of the map $\beta$.}

\change{34}{-14}{$\mu=\beta \circ \hat \eta\circ \beta^{-1}$}{$\mu=\beta \circ \hat \eta\circ \beta^{-1}$ up to the framing term}

\compl{35}{2}{The orientations of the closed components have not been indicated.}

\change{35}{-2}{follows from}{follow from}

\change{36}{1}{Recall from 5.4}{Recall from Remark 5.4}

\compl{39}{1}{It might be better not to call $\beta$ the straight vertical strand, since the notation $\beta$
is already used intensively for something else.}

\change{39}{-11}{$Z(\Phi_1)$}{$Z(\Phi)$}

\change{39}{-4}{$\tilde{\mathcal{A}}/2$}{$\tilde{\mathcal{A}}^{/2}$}

\change{41}{-13}{satatement}{statement}

\change{41}{-6}{$\hbox{gr}_a \lambda$}{$\hbox{gr} \lambda_a$}

\change{42}{-1}{$\lambda^{alg}_1$}{$\lambda^{alg}_a$}

\end{document}