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\title{List of edits for ``Finite type invariants of w-knotted objects: tangles, foams and the Kashiwara-Vergne problem''}
\author{Dror Bar-Natan and Zsuzsanna Dancso}
\begin{document}
\maketitle
We would like to thank the referee for pointing out several ways to improve clarity and make the paper easier to read, and we agree with all of
the points raised. We made changes
to implement these suggestions while aiming not to increase the overall length of the paper. Below is a list of changes.
\begin{enumerate}
\item We added a section (Section 1.2) to the introduction, which serves as a general overview of the paper and contains a large scale
explanation of the results. In particular, we list and explain the three main theorems and mention the tools we use. We also explain the
precise relation between the Reidemeister theory (circuit algebra presentation) of w-tangles and w-foams and 4-dimensional topology.
(Indeed there is a surjective map which is conjectured to be injective.)
\item The addition of the section above made it unnecessary to list the section summaries in the introduction, hence we deleted these.
\item Added thanks to the referee in acknowledgements.
\item In Section 2 we added a simple running example (namely, $\mathbb Z$) to illustrate the notions of projectivizations and expansions.
We felt that the proof of the projectivization result for quandles (previously included in the ``Odds and Ends'' section) no longer adds
to the readability of the paper but adds to the length, so we replaced it with a reference only.
\item Regarding circuit algebras, the prime example (v- and w- tangles) is presented at the beginning of Section 3, right after the
the definition of circuit algebras in Section 2. We don't think that the reader would gain a lot from adding a different example in Section 2;
however, we do agree that the utility of circuit algebras (namely, a significant simplification of the presentation of v- and w- tangles)
was not sufficiently clear in the previous preprint. To correct this, we added a paragraph (2nd paragraph of Section 2.4) explaining this point;
to readers already familiar with virtual tangles this will also provide an example to keep in mind. We also moved up and extended Remark 3.2
to explain virtual crossings (or the lack thereof) in Definition 3.1 and to clarify the corresponding figure.
\item We corrected the typo in the definition of $\delta$, as well as some other minor errors that we have noticed while editing.
\item In Section 4, we added several ``summary definitions'' for the spaces we are working with so that the reader doesn't have to flip
trough the pages to collect the imposed relations and operations. For $wTF^o$ the summary is on page 34, for its projectivization see
Definition 4.8 on page 36, for $wTF$ and its projectivization see page 42. We also made the definition of $sKTG$ more explicit in Definition 4.12
on page 45.
\end{enumerate}
\end{document}