\documentclass[10pt,notitlepage]{article}
\usepackage{amsmath,graphicx,amssymb,color,datetime,dbnsymb,stmaryrd,mathtools}
\usepackage[setpagesize=false]{hyperref}

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\paperheight 11in
\textwidth 8.5in
\textheight 11in
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\newcounter{linecounter}
\newcommand{\cheatline}{\bf\refstepcounter{linecounter}\thelinecounter. }
\def\checkmarkb{{$\checkmark_{\!\!b}\,$}}

\def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}}

\def\calA{{\mathcal A}}
\def\red{\color{red}}

\def\decoratedV#1#2#3{{
  \prescript{}{\text{\tiny #2}}{
    \overset{\text{\tiny\!\!#1\!\!}}{\curlywedgeuparrow}
  }_\text{\tiny #3}
}}

\def\navigator{{Dror Bar-Natan: Pensieve: 2013-11: DoubleTree @ \today, \ampmtime}}

\def\talkurl{{\url{http://drorbn.net/AcademicPensieve/2013-11/DoubleTree/}}}

\def\Wisdom{{\raisebox{0mm}{\parbox[t]{4in}{
{\red Random Wisdom.}
\newline$\bullet$~In double trees, the punctured side is always above the
capped side.
}}}}

\def\GeneralEqns{{\raisebox{0mm}{\parbox[t]{3.95in}{
{\red General Equations.}\hfill{\footnotesize
  All verifications at {\tt DoubleTreeVerification.nb}%
}
\setcounter{linecounter}{0}
\newline{\checkmark\cheatline R4}\hfill $R_{23}R_{13}V=VR_{12,3}$
\newline{\checkmark\cheatline Twist}\hfill $V\Theta_{12}=RV^{21}$
\newline{\checkmark\cheatline NonDeg}\hfill $V\act d\eta^1=V\act d\eta^2=1$
\newline{\checkmark\cheatline Cup}\hfill $C=C\act dS$
\newline{\checkmark\cheatline $\Phi$}\hfill
  $\Phi\coloneqq V_{12,3}^{-1}V^{-1}V_{23}V_{1,23}$
\newline{\checkmark\cheatline $\pentagon$}\hfill
  $\Phi\Phi_{1,23,4}\Phi_{234}=\Phi_{12,3,4}\Phi_{1,2,34}$
\newline{\cheatline $\hexagon_\pm$}\hfill
  $\Theta_{12,3}^\pm
    =\Phi\Theta_{23}^\pm\Phi_{132}^{-1}\Theta_{13}^\pm\Phi_{312}$
\newline{\checkmark\cheatline HF4$\Phi$}\hfill $\Phi\Phi_{321}=1$
\newline{\checkmark\cheatline VF4$\Phi$}\hfill $\Phi(\Phi\act dS^{123})=1$
\newline{\checkmark\cheatline OH}\hfill
  $\Phi_{01,2,3}\act dS^{23}\act dm^{03}_0\act dm^{12}_1=1$
\newline{\checkmark\cheatline $\nu$}\hfill
  $\nu\coloneqq(\Phi\act dS^2 \act dm^{321}_1)^{-1}$
\newline{\checkmark\cheatline $C$}\hfill $C^4=\nu$
\newline{\checkmark\cheatline $\decoratedV{1}{cup}{pun}$}\hfill
  $V\act h\eta^1\act t\eta^2\act dm^{21}_1 = R_{11}^{1/2}$
\newline{\checkmark\checkmarkb\cheatline $\decoratedV{1}{pun}{cup}$}\hfill
  $V\act h\eta^2\act t\eta^1\act dm^{12}_1 = 1$
\newline{\cheatline $\decoratedV{cap}{1}{pun}$}\hfill
  $V\act t\eta^2\act dS^1\act dm^{21}_1 = R_{11}^{-1/2}$
\newline{\checkmark\cheatline $\decoratedV{cap}{pun}{1}$}\hfill
  $V\act t\eta^1\act dS^2\act hm^{12}_1 = 1$
\newline{\checkmark\cheatline The Buckle}\hfill $Z_\otimes\coloneqq
  \Phi^{-1}_{13,2,4}\Phi_{132}\Theta_{32}\Phi^{-1} \Phi_{12,3,4}$
\newline{\checkmark\cheatline Buckle2$V$}\hfill $V_\otimes\coloneqq
  Z_\otimes\act t\eta^1 h\eta^2 \act dm^{12}_1
  \act t\eta^3 h\eta^4\act dm^{34}_2$
}}}}

\def\UnitaryVEqns{{\raisebox{0mm}{\parbox[t]{4in}{
\setcounter{linecounter}{0}
{\red Equations for Vanilla / Unitary $V$.}
\newline{\checkmark\cheatline Unitarity}\hfill $V(V\act dA^{12})=1$
\newline{\checkmark\cheatline VF4V}\hfill$V(V\act dS^{12})=R$
\newline{\checkmark\cheatline Cap}\hfill $VC_{12}=C_1C_2$
  in $\calA(\cappededge_{12})$
\newline{\cheatline $\decoratedV{del}{1}{$S$}$}\hfill 
  $V\act dS^2\act dm^{12}_1 = C^{-2}(R_{11}^{-1/2}\act dS^1)$
\newline{\cheatline $\decoratedV{pun}{1}{cup}$}\hfill
  $V\act h\eta^2\act dS^1\act dm^{21}_1 = C^{-2}$
\newline{\cheatline $\decoratedV{pun}{cup}{1}$}\hfill
  $V\act h\eta^1\act dS^2 \act dm^{12}_1=((R_{11}^{1/2}\act dS^1)C^2)^{-1}$
\newline{\checkmark\cheatline $VV_\otimes$}\hfill
  $V=C_1^{-1}C_2^{-1}V_\otimes C_{12}$
}}}}

\def\SimpleBuckleEqns{{\raisebox{0mm}{\parbox[t]{4in}{
\setcounter{linecounter}{-1}
{\red Equations with Simple Buckle.}\hfill
  {\checkmarkb\cheatline}$V_b\coloneqq C_1C_2VC^{-1}_{12}$
\newline{\checkmarkb\cheatline Unitarity}\hfill $V_bC^2_{12}(V_b\act dA^{12})=C^2_1C^2_2$
\newline{\checkmarkb\cheatline VF4V}\hfill$V_bC^2_{12}(V_b\act dS^{12})=RC^2_1C^2_2$
\newline{\checkmarkb\cheatline Cap}\hfill $V_bC^2_{12}=C^2_1C^2_2$ in $\calA(\cappededge_{12})$
\newline{\cheatline $\decoratedV{del}{1}{$S$}$}\hfill 
  $V_b\act dS^1\act dm^{21}_1 = R_{11}^{-1/2}$
\newline{\cheatline $\decoratedV{pun}{1}{cup}$}\hfill
  $V_b\act h\eta^2\act dS^1\act dm^{21}_1 = 1$
\newline{\cheatline $\decoratedV{pun}{cup}{1}$}\hfill
  $V_b\act h\eta^1\act dS^2 \act dm^{12}_1=(R_{11}^{1/2}\act dS^1)^{-1}$
\newline{\checkmarkb\cheatline $V_bV_\otimes$}\hfill $V_b=V_\otimes$
}}}}

\def\SimpleCapEqns{{\raisebox{0mm}{\parbox[t]{4in}{
\setcounter{linecounter}{-1}
{\red Equations with Trivial Caps.}\hfill{\cheatline}$V_c\coloneqq C_{12}VC_1^{-1}C_2^{-1}$
\newline{\cheatline Unitarity}\hfill $V_cC^{-2}_{12}(V_c\act dA^{12})=C^{-2}_1C^{-2}_2$
\newline{\cheatline VF4V}\hfill$V_cC^{-2}_{12}(V_c\act dS^{12})=RC^{-2}_1C^{-2}_2$
\newline{\cheatline Cap}\hfill $V_c=1$ in $\calA(\cappededge_{12})$
\newline{\cheatline $\decoratedV{del}{1}{$S$}$}\hfill 
  $V_c\act dS^2\act dm^{12}_1 = C^{-4}(R_{11}^{-1/2}\act dS^1)$
\newline{\cheatline $\decoratedV{pun}{1}{cup}$}\hfill
  $V_c\act h\eta^2\act dS^1\act dm^{21}_1 = C^{-4}$
\newline{\cheatline $\decoratedV{pun}{cup}{1}$}\hfill
  $V_c\act h\eta^1\act dS^2 \act dm^{12}_1=((R_{11}^{1/2}\act dS^1)C^4)^{-1}$
\newline{\cheatline $V_cV_\otimes$}\hfill $V_c=C_1^{-2}C_2^{-2}C^2_{12}V_\otimes$
}}}}

\def\ToDo{{\raisebox{0mm}{\parbox[t]{4in}{
{\bf To Do.}
$\bullet$~The double tree procedure $\doubletree$.
$\bullet$~Well-definededness.
$\bullet$~$\doubletree=\doubletree\doubletree$?
$\bullet$~Behaviour under disjoint unions.
$\bullet$~$\doubletree(\uparrow)$.
$\bullet$~Behaviour under strand concatanation.
$\bullet$~Behaviour under annulus unzip.
$\bullet$~$\doubletree(\upcap)$.
$\bullet$~$\doubletree(\upcap\upcap)=\doubletree(\upcap)\doubletree(\upcap)$?
$\bullet$~Behaviour under disk unzip.
}}}}

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