===== recycled on Wed Jun 25 12:30:27 EDT 2014 by drorbn on Debian-1207 ====== Given that $\delta\colon\calA^w(\uparrow_S)\to\calA^w(S;S)$ is an isomorphism for every $S$, we can regard the operations of defined on the $\calA^w(\uparrow_S)$-spaces in Definition~\ref{def:Operations} as operations on the system of spaces $\{\calA^w(S;S)\}$. The next proposition shows how these operations can be written in terms of the head and tail'' operations of Definition~\ref{def:AHTOperations}, thus completing the description of the $E_s$ presentation. ===== recycled on Wed Jul 2 14:29:48 EDT 2014 by drorbn on debian-1405 ====== \noindent\parbox{3.5in}{ \item $d\eta^s\act\delta = \delta\act h\eta^s\act t\eta^s$. \item \label{it:HTdA} $dA^s \act \delta = \delta \act hA^s \act tA^s\act tha^{ss}$. \item \label{it:HTdS} $dS^s \act \delta = \delta \act hS^s \act tS^s\act tha^{ss}$. }\hfill\parbox{3in}{ \item \label{it:HTdm} $dm^{ab}_c \act \delta = \delta \act tha^{ab} \act hm^{ab}_c \act tm^{ab}_c$. \item $d\Delta^a_{bc} \act \delta = \delta \act h\Delta^a_{bc} \act t\Delta^a_{bc}$. \item $d\sigma^a_b \act \delta = \delta \act h\sigma^a_b \act t\sigma^a_b$. } ===== recycled on Mon Sep 22 17:38:35 EDT 2014 by drorbn on Debian-1207 ====== In other words, given $\lambda=\{s\to\lambda_s\}_{s\in S}\in\FL(S)^S$ and $\omega\in\CW(S)$, we wish to find $\lambda'$ and $\omega'$ such that $E_l(\lambda;\,\omega)=E_f(\lambda';\,\omega')$. Given $(\lambda;\,\omega)$ as above and a scalar $t$, let $\Gamma(\lambda,t)=\{s\to\gamma_s(t)\}\in\FL(S)^S$ be the unique solution of the system of ordinary differential equations $$\label{eq:Gamma} \forall s\in S,\quad \frac{d\gamma_s(t)}{dt} = \gamma_s(t)\act e^{-t\partial_\lambda} \act\frac{\ad\gamma_s(t)}{e^{\ad\gamma_s(t)}-1}; \qquad \gamma_s(0)=0.$$ Let $\Gamma(\lambda)\coloneqq\Gamma(\lambda,1)$. ===== recycled on Thu Oct 2 18:32:14 EDT 2014 by drorbn on Debian-1207 ====== \footnotetextC{In computer talk: \shortdialoginclude{bch} } \footnotetextC{Just to show that we can, here are the lexicographically middle three of the 2,181 terms of the BCH series in degree 16, along with the time in seconds it took my humble laptop tocompute it: \shortdialoginclude{bch16} } ===== recycled on Mon Oct 13 15:25:11 EDT 2014 by drorbn on Debian-1207 ====== \def\FLQuote#1{{% \newline\vspace{0mm} {\hspace{-1mm}\imagetop{\includegraphics[width=6mm]{figs/FreeLie.eps}}\ \imagetop{\includegraphics[scale=0.16]{ComputerTalk/FL#1.eps}} } \newline\vskip 0mm }} ===== recycled on Wed Jan 28 19:59:24 EST 2015 by drorbn on Debian-1207 ====== We can present/compute that invariant (more precisely, its logarithm) using either the $\TW_l(S)$-valued \cite{AT}-presentation $E_l$ or using the $\TW_s(S)$-valued factored presentation $E_f$ (recall Figure~\ref{fig:diagram}). Let $\zeta_l$ and $\zeta_s$ be these two presentations. Namely, $\zeta_l$ and $\zeta_s$ are defined by $Z=\zeta_l\act E_l \qquad\text{and}\qquad Z=\zeta_s\act E_s\act \delta^{-1}=\zeta_s\act E_f.$ ===== recycled on Fri Jul 3 21:44:55 EDT 2015 by drorbn on Debian-1207 ====== $\partial(0, \ldots, \smallunderbrace{x_j}_i, \ldots, \smallunderbrace{x_i}_j, \ldots, 0)$. ===== recycled on Sun Sep 6 20:12:53 EDT 2015 by drorbn on Debian-1207 ====== \section{Introduction} This paper being a third in a series~\cite{WKO1,WKO2}, as well as a continuation of~\cite{AT, AlekseevEnriquezTorossian:ExplicitSolutions} and of~\cite{KBH}, we will forgo a description of the context and the motivations and forgo the precise definitions, and instead jump right into the heart of the matter --- the equations we seek to solve, and the spaces in which they are written. Our fundamental quantities are \begin{itemize} \item $R=\exp(\rightarrowdiagram)$, the $Z^w$-value of a crossing, a member of the space $\calA^w(\uparrow_2)$ defined in~\cite{WKO1} and reviewed in Section~\ref{subsec:Spaces} below. \item $V$, the $Z^w$-value of a vertex, a member of $\calA^w(\uparrow_3)$. \item $C\in\calA^w(\upcap)$, the $Z^w$-value of a cap. \item A Drinfel'd associator $\Phi$ and a braiding element for u-braids $\Theta=\exp(\frac12\horizontalchord)$. \end{itemize} \subsection{The Equations} \label{subsec:Equations} \begin{itemize} \item Reidemeister 4, R4 \hfill$\displaystyle R_{23}R_{13}V=VR_{12,3}$ \quad(\stepcounter{equation}\theequation) \end{itemize} \subsection{The Spaces} \label{subsec:Spaces} ===== recycled on Tue Sep 15 19:20:12 EDT 2015 by drorbn on Debian-1207 ====== \begin{proof} The two sides $L_t$ and $R_t$ of Equation~\eqref{eq:Gamma} are power-series perturbations of the identity automorphism of $\FL(S)$. More fully, $L_t$ can be written $L_t=\sum_{d\geq 0}L_t(d)$ where $L_t(d)\colon\FL(S)\to\FL(S)$ raises degrees by exactly $d$ and depends on $t$ as a polynomial of degree at most $d$, and where $L_t(0)$ is the identity for all $t$, $L_0(d)$ is $0$ for all positive $d$, and $R_t$ can be written in a similar way. We claim that it is enough to prove that $(\frac{dL_t}{dt})\act L_t^{-1} = (\frac{dR_t}{dt})\act R_t^{-1}$. Indeed, if $L_t\neq R_t$, consider the minimal degree $k$ in the variable $t$ in which they differ. Then $k>0$ ===== recycled on Thu Oct 8 19:48:54 EDT 2015 by drorbn on Debian-1207 ====== Namely, if $D_1,D_2\in\calA^w_{\exp}(\uparrow_S)$, $\pi_\fraka(D_1)=\pi_\fraka(D_2)$, $C_{D_1}=C_{D_2}$, and $\downcap(D_1)=\downcap(D_2)$, then $D_1=D_2$.