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\newlabel{fig:ZuDelete}{{26}{33}{In \cite [Section 4.6.1]{Bar-NatanDancso:WKO2} $Z^u$ is constructed from an invariant $Z^{old}$ by applying vertex normalizations, which depend on vertex signs: these are shown along the top horizontal arrow of each diagram (see also \cite [Figure 29]{Bar-NatanDancso:WKO2}). It follows $Z^u$ is only homomorphic up to a correction term when deleting the top edge of a positive vertex (first in the total ordering around the vertex) or the bottom edge of a negative vertex: see the top two diagrams. In other edge deletions the normalizations cancel, and hence $Z^u$ is homomorphic with respect to these edge deletions, as for example in the bottom two diagrams}{figure.26}{}}
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\newlabel{fig:Square7Comm}{{31}{37}{Computing the top left corner of Square 7, Step 1: $\doubletree (T)$ can be expressed as the $\sKTG $ denoted $S$ inserted into the $\sKTG $ denoted $A$, followed by unzips, as shown. $Z^{u}$ respects insertions, hence computing $Z^{u}(A)$ determines the value of $Z^{u}(\doubletree (T))$ outside of $S$}{figure.31}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {32}{\ignorespaces Computing the top left corner of Square 7, Step 2: computing $Z^{u}(A)$. The ${\mathit  s\!K\!T\!G}$ $A$ can be obtained by inserting the {\em  buckle} ${\mathit  s\!K\!T\!G}$ twice into a simpler ${\mathit  s\!K\!T\!G}$, and unzipping, as shown on the left. The value of the buckle was computed in Figure \ref  {fig:BuckleBraid}. Using this value---denoted $\beta ^{u}$---and the algorithm in \cite  [Section 5.2]{Bar-NatanDancso:WKO2}, one computes $Z^{u}(A)$. The result is denoted $D$ and shown on the right.}}{37}{figure.32}\protected@file@percent }
\newlabel{fig:Square7Comm2}{{32}{37}{Computing the top left corner of Square 7, Step 2: computing $Z^{u}(A)$. The $\sKTG $ $A$ can be obtained by inserting the {\em buckle} $\sKTG $ twice into a simpler $\sKTG $, and unzipping, as shown on the left. The value of the buckle was computed in Figure \ref {fig:BuckleBraid}. Using this value---denoted $\beta ^{u}$---and the algorithm in \cite [Section 5.2]{Bar-NatanDancso:WKO2}, one computes $Z^{u}(A)$. The result is denoted $D$ and shown on the right}{figure.32}{}}
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\newlabel{fig:DTandBuckle}{{37}{41}{On the left we show how to obtain $\doubletree (\FlippedYGraph )$ via an unzip from $B^u$ insterted into $\doubletree (\curvearrowright )$. From this we compute $\xi (\FlippedYGraph )$, and finally $V$ on the right}{figure.37}{}}
\newlabel{cor:SameResult}{{3.20}{41}{}{theorem.3.20}{}}
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\newlabel{fig:ValueV}{{39}{44}{To compute $\varphi (\Phi ^{-1}(a_{2(13)}, -a_{2(13)}-a_{4(13)}) \cdot e^{a_{23}/2} \cdot \Phi (a_{23},a_{43}))$ we switch to a {\em placement notation} in which we mark on each skeleton strand the elements that have arrows ending on it. For this purpose we denote $\Phi ^{-1}(a_{2(13)}, -a_{2(13)}-a_{4(13)})=:\psi $ and $\Phi (a_{23},a_{43})=:\chi $}{figure.39}{}}
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\bibcite{BDV:OU}{BDV}
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\bibcite{Drinfeld:QuasiHopf}{Dr}
\bibcite{Jones:PlanarAlgebras}{J}
\bibcite{KashiwaraVergne:Conjecture}{KV}
\bibcite{LeMurakami:Universal}{LM}
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