In Section 4 of ~\cite{WKO2} we introduce and study {\em w-tangled
foams}.  These are defined combinatorially, as a finitely generated {\em
circuit algebra} with certain Reidemeister relations. For motivation, we
present a local topological interpretation of w-tangled foams as tangled
tubes in $\bbR^4$.  Unfortunately, some errors have occurred in this
interpretation, stemming from our lack of care around ``2D orientations'',
as below. We wish to thank Yusuke Kuno and Haruko Miyazawa for noting
these issues.

\begin{itemize}
\parpic[r]{\input{figs/R4Issue.\figsuffix}}
\item If foams are to have 2D orientations, they cannot be glued to
each other unless these orientations match.  This breaks the circuit
algebra structure. This can be corrected by switching to {\em coloured}
circuit algebras, in which strands and strand-ends are coloured by
their 2D orientations, and gluings are allowed only if colours match. However, we
choose a different resolution explained below.
\item We missed that the ``$A_e$'' operations of flipping only the 1D but not the 2D
orientations of a strand interact unpleasantly with R4 moves, as shown
on the right. This can be corrected by giving more care to the (2D)
colours of the edges next to a vertex and making the precise meaning of
the R4 move depend on these colours. Since this is notationally tedious, we choose 
a different resolution below.
\end{itemize}

Although the resolution is a small change to the content of the paper, it
necessitates some notation and language changes, and for the convenience
of our readers we have incorporated these into the body of the paper as
a revised version, available at~\cite{WKO2v2}.

\parpic[r]{\input{figs/WenShort.\figsuffix}}
The easier solution to both of the problems above is to forgo of
2D orientations and of the operations $A_e$ altogether, and to
replace them with more serious attention to the {\em wens}, which are cut open Klein-bottles 
(\cite[Section~4.5]{WKO2}, and \cite[Section~4.1]{WKO2v2}). Indeed, conjugating a 
strand by a wen and reversing it (strand reversal is the operation $S_e$, unchanged from the original 
paper) has the same effect on crossings as the operation $A_e$ 
was meant to have. 

The main result of the paper is identifying {\em homomorphic expansions} for w-tangled foams with 
solutions to the Kashiwara-Vergne (KV) equations. A crucial equation relating the values of such an expansion, the {\em unitarity equation} $V\cdot
A_1A_2(V)=1$ \cite[Equation~(12)]{WKO2} is replaced by an equivalent equation,
$V^*V=1$ \cite[Equation~(U), page 41]{WKO2v2} in which $V^*$ is obtained from $V$
by multiplying it by a wen on all three sides. 

In \cite{WKO2} we claimed that the value of the wen under an expansion must be 1, and that expansions for w-tangled foams are in bijection with Kashiwara-Vergne solutions with {\em even Duflo function}. In fact, the value of the wen could be non-trivial \cite[Lemma 4.9]{WKO2v2}, but the statement regarding KV solutions remains true for homomorphic expansions where the value of the wen is set to be $1$. 

Theorem~4.9 of \cite{WKO2} stated that for ``orientable w-tangled foams'', i.e. where wens are not included, homomorphic expansions are in one to one correspondence with KV solutions in general. This theorem remains true: in \cite[Section 4.6]{WKO2v2} we present a more careful definition of orientable w-tangled foams as a sub-circuit algebra. Edge-wise adjoint operations replaced by the well-defined global orientation reversal and composition with wens at every tangle as above, noting that these wens all cancel. Homomorphic expansions for these foams are indeed in one to one correspondence with KV solutions.

We note that the results of papers \cite{BDS, DHR} which build on Section 4 of \cite{WKO2} are unaffected by this Corrigendum: in these papers one can simply mechanically replace $A_1A_2(V)$ with $V^*$.

\begin{thebibliography}{BND1}

\bibitem[BND]{WKO2} D.~Bar-Natan and Z.~Dancso,
    {\em Finite Type Invariants of W-Knotted Objects II: Tangles and
      the Kashiwara-Vergne Problem,}
  Math.\ Ann.\ {\bf 367} (2017) 1517--1586,
  \arXiv{1405.1955v3}.

\bibitem[BNDv2]{WKO2v2} D.~Bar-Natan and Z.~Dancso,
  \href
    {http://drorbn.net/AcademicPensieve/Projects/WKO2}
    {{\em Finite Type Invariants of W-Knotted Objects II: Tangles and
      the Kashiwara-Vergne Problem,}}
  \url{http://drorbn.net/WKO2} and \arXiv{1405.1955}.
  
\bibitem[BDS]{BDS} D.~Bar-Natan, Z.~Dancso, N.~Scherich,
{\em Ribbon 2-knots, $1+1=2$, and Duflo's Theorem for Arbitrary Lie Algebras}
Alg.\ Geom.\ Topol.\  {\bf 20}(7) 3733--3760. \arXiv{1811.08558}

  
\bibitem[DHR]{DHR} Z.~Dancso, I.~Halacheva, M.~Robertson, 
{\em A Topological Characterisation of the Kashiwara--Vergne Groups}
Trans.\ Amer.\ Math.\ Soc. {\bf 376}(5), 3265--3317. \arXiv{2106.02373}

\end{thebibliography}
