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The third step uses that the Kontsevich integral of KTGs is homomorphic with respect to the ``connected sum'' operation and that the value of the unknot is $\nu $ (see \cite {Bar-NatanDancso:KTG} for an explanation of both of these facts)}{figure.23}{}} \@writefile{lof}{\contentsline {figure}{\numberline {24}{\ignorespaces Normalizations for $Z^u$ at the vertices.}}{52}{figure.24}\protected@file@percent } \newlabel{fig:ZatVertices}{{24}{52}{Normalizations for $Z^u$ at the vertices}{figure.24}{}} \newlabel{subsec:wTFcompatibility}{{4.8}{52}{The relationship between $\sKTG $ and $\wTF $}{subsection.4.8}{}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{4.8}{The relationship between ${\mathit s\!K\!T\!G}$ and ${\mathit w\!T\!F}$}}{52}{subsection.4.8}\protected@file@percent } \newlabel{eq:uwcompatibility}{{15}{52}{The relationship between $\sKTG $ and $\wTF $}{equation.4.15}{}} \newlabel{thm:ZuwCompatible}{{4.28}{52}{}{theorem.4.28}{}} \citation{AlekseevTorossian:KashiwaraVergne} \citation{AlekseevTorossian:KashiwaraVergne} \citation{AlekseevTorossian:KashiwaraVergne} \newlabel{lem:TreesAndUnitarity}{{4.29}{53}{}{theorem.4.29}{}} \newlabel{eq:twist}{{16}{53}{The relationship between $\sKTG $ and $\wTF $}{equation.4.16}{}} \citation{AlekseevTorossian:KashiwaraVergne} \citation{AlekseevTorossian:KashiwaraVergne} \citation{AlekseevTorossian:KashiwaraVergne} \@writefile{lof}{\contentsline {figure}{\numberline {25}{\ignorespaces The $Z^w$-value of the right associator.}}{54}{figure.25}\protected@file@percent } \newlabel{fig:wAssociator}{{25}{54}{The $Z^w$-value of the right associator}{figure.25}{}} \newlabel{eq:TwistWithF}{{17}{54}{The relationship between $\sKTG $ and $\wTF $}{equation.4.17}{}} \newlabel{eq:AssociatorAndV}{{18}{54}{The relationship between $\sKTG $ and $\wTF $}{equation.4.18}{}} \citation{AlekseevTorossian:KashiwaraVergne} \citation{AlekseevTorossian:KashiwaraVergne} \citation{AlekseevTorossian:KashiwaraVergne} \citation{Drinfeld:QuasiHopf} \citation{Bar-NatanDancso:KTG} \citation{AlekseevTorossian:KashiwaraVergne} \citation{AlekseevTorossian:KashiwaraVergne} \citation{AlekseevTorossian:KashiwaraVergne} \newlabel{eq:ATPhiandV}{{19}{55}{The relationship between $\sKTG $ and $\wTF $}{equation.4.19}{}} \citation{Bar-NatanDancso:WKO1} \@writefile{lof}{\contentsline {figure}{\numberline {26}{\ignorespaces The proof of Equation (\ref {eq:NooseCapped}). Note that the unzips are ``illegal'', as the strand directions don't match. This can be fixed by inserting a small bubble at the bottom of the noose and doing a number of orientation switches. As this doesn't change the result or the main argument, we suppress the issue for simplicity. Equation (\ref {eq:NooseCapped}) is obtained from this result by multiplying by $S(C)^{-1}$ on the bottom and by $C^{-1}$ on the top.}}{56}{figure.26}\protected@file@percent } \newlabel{fig:NooseCappedProof}{{26}{56}{The proof of Equation (\ref {eq:NooseCapped}). Note that the unzips are ``illegal'', as the strand directions don't match. This can be fixed by inserting a small bubble at the bottom of the noose and doing a number of orientation switches. As this doesn't change the result or the main argument, we suppress the issue for simplicity. Equation (\ref {eq:NooseCapped}) is obtained from this result by multiplying by $S(C)^{-1}$ on the bottom and by $C^{-1}$ on the top}{figure.26}{}} \newlabel{eq:NooseSymmetry}{{20}{56}{The relationship between $\sKTG $ and $\wTF $}{equation.4.20}{}} \newlabel{eq:NooseCapped}{{21}{56}{The relationship between $\sKTG $ and $\wTF $}{equation.4.21}{}} \newlabel{prop:uwBT}{{4.31}{57}{}{theorem.4.31}{}} \@writefile{toc}{\contentsline {section}{\tocsection {}{5}{Odds and Ends}}{57}{section.5}\protected@file@percent } \newlabel{sec:odds}{{5}{57}{Odds and Ends}{section.5}{}} \newlabel{subsec:CAMotivation}{{5.1}{57}{Motivation for circuit algebras: electronic circuits}{subsection.5.1}{}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.1}{Motivation for circuit algebras: electronic circuits}}{57}{subsection.5.1}\protected@file@percent } \@writefile{lof}{\contentsline {figure}{\numberline {27}{\ignorespaces The J-K flip flop, a very basic memory cell, is an electronic circuit that can be realized using 9 components --- two triple-input ``and'' gates, two standard ``nor'' gates, and 5 ``junctions'' in which 3 wires connect (many engineers would not consider the junctions to be real components, but we do). Note that the ``crossing'' in the middle of the figure is merely a projection artifact and does not indicate an electrical connection, and that electronically speaking, we need not specify how this crossing may be implemented in ${\mathbb R}^3$. The J-K flip flop has 5 external connections (labelled J, K, CP, Q, and Q') and hence in the circuit algebra of computer parts, it lives in $C_5$. In the directed circuit algebra of computer parts it would be in $C_{3,2}$ as it has 3 incoming wires (J, CP, and K) and two outgoing wires (Q and Q'). }}{57}{figure.27}\protected@file@percent } \newlabel{fig:FlipFlop}{{27}{57}{The J-K flip flop, a very basic memory cell, is an electronic circuit that can be realized using 9 components --- two triple-input ``and'' gates, two standard ``nor'' gates, and 5 ``junctions'' in which 3 wires connect (many engineers would not consider the junctions to be real components, but we do). Note that the ``crossing'' in the middle of the figure is merely a projection artifact and does not indicate an electrical connection, and that electronically speaking, we need not specify how this crossing may be implemented in $\bbR ^3$. The J-K flip flop has 5 external connections (labelled J, K, CP, Q, and Q') and hence in the circuit algebra of computer parts, it lives in $C_5$. In the directed circuit algebra of computer parts it would be in $C_{3,2}$ as it has 3 incoming wires (J, CP, and K) and two outgoing wires (Q and Q')}{figure.27}{}} \@writefile{lof}{\contentsline {figure}{\numberline {28}{\ignorespaces The circuit algebra product of 4 big black components and 1 small black component carried out using a green wiring diagram, is an even bigger component that has many golden connections (at bottom). When plugged into a yet bigger circuit, the CPU board of a laptop, our circuit functions as 4,294,967,296 binary memory cells. }}{58}{figure.28}\protected@file@percent } \newlabel{fig:Circuit}{{28}{58}{The circuit algebra product of 4 big black components and 1 small black component carried out using a green wiring diagram, is an even bigger component that has many golden connections (at bottom). When plugged into a yet bigger circuit, the CPU board of a laptop, our circuit functions as 4,294,967,296 binary memory cells}{figure.28}{}} \newlabel{subsec:sKTGgensProof}{{5.2}{58}{Proof of Proposition \ref {prop:sKTGgens}}{subsection.5.2}{}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.2}{Proof of Proposition \ref {prop:sKTGgens}}}{58}{subsection.5.2}\protected@file@percent } \citation{AlekseevTorossian:KashiwaraVergne} \@writefile{toc}{\contentsline {section}{\tocsection {}{6}{Glossary of notation}}{60}{section.6}\protected@file@percent } \newlabel{sec:glossary}{{6}{60}{Glossary of notation}{section.6}{}} \bibcite{AlekseevMeinrenken:KV}{AM} \bibcite{AlekseevTorossian:KashiwaraVergne}{AT} \bibcite{AlekseevEnriquezTorossian:ExplicitSolutions}{AET} \bibcite{Bar-Natan:OnVassiliev}{BN1} \bibcite{Bar-Natan:NAT}{BN2} \bibcite{Bar-Natan:Associators}{BN3} \bibcite{Bar-Natan:AKT-CFA}{BN4} \bibcite{Bar-NatanDancso:KTG}{BND} \bibcite{Bar-NatanGaroufalidisRozanskyThurston:WheelsWheeling}{BGRT} \bibcite{Bar-NatanHalachevaLeungRoukema:v-Dims}{BHLR} \bibcite{Bar-NatanLeThurston:TwoApplications}{BLT} \bibcite{BerceanuPapadima:BraidPermutation}{BP} \bibcite{BrendleHatcher:RingsAndWickets}{BH} \bibcite{ChepteaLe:EvenAssociator}{CL} \bibcite{CarterSaito:KnottedSurfaces}{CS} \bibcite{Dahm:GeneralBraid}{D} \bibcite{Dancso:KIforKTG}{Da} \bibcite{Drinfeld:QuantumGroups}{Dr1} \bibcite{Drinfeld:QuasiHopf}{Dr2} \bibcite{Drinfeld:GalQQ}{Dr3} \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{61}{section*.6}\protected@file@percent } \bibcite{EtingofKazhdan:BialgebrasI}{EK} \bibcite{FennRimanyiRourke:BraidPermutation}{FRR} \bibcite{Goldsmith:MotionGroups}{Gol} \bibcite{Jones:PlanarAlgebrasI}{Jon} \bibcite{KashiwaraVergne:Conjecture}{KV} \bibcite{Kauffman:VirtualKnotTheory}{Ka} \bibcite{Kuperberg:VirtualLink}{Kup} \bibcite{LeMurakami:Universal}{LM} \bibcite{Leinster:Higher}{Lei} \bibcite{Levine:Addendum}{Lev} \bibcite{Loday:LeibnizAlg}{Lod} \bibcite{McCool:BasisConjugating}{Mc} \bibcite{MilnorMoore:Hopf}{MM} \bibcite{MurakamiOhtsuki:KTGs}{MO} \bibcite{Satoh:RibbonTorusKnots}{Sa} \bibcite{WKO}{WKO0} \bibcite{Bar-NatanDancso:WKO1}{WKO1} \bibcite{Bar-NatanDancso:WKO3}{WKO3} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{15.01373pt} \newlabel{tocindent1}{20.88867pt} \newlabel{tocindent2}{34.5pt} \newlabel{tocindent3}{0pt} \gdef \@abspage@last{62}