\documentclass[11pt,notitlepage]{article}
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\def\sheeturl{{\url{http://drorbn.net/AcademicPensieve/2013-04/}}}

\def\tbd{\text{\color{red} ?}}

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\begin{document}
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{\LARGE\bf Cheat Sheet $\beta$}\hfill
\parbox[b]{5.2in}{\small
  \null\hfill\sheeturl
  \newline\null\hfill initiated 24/3/13; continues \href{http://drorbn.net/AcademicPensieve/2013-03/}{2013-03}; continued \href{http://drorbn.net/AcademicPensieve/2013-05/}{2013-05}; modified \today, \ampmtime
}

\rule{\textwidth}{1pt}

The original $\beta$-calculus: With $\epsilon:=1+\alpha$, $\langle\alpha\rangle:=\sum_v\alpha_v$,
and $\langle\gamma\rangle:=\sum_{v\neq u}\gamma_v$,

\vskip 1mm  $\displaystyle
  \begin{array}{c|c}\omega_1&H_1\\\hline T_1&A_1\end{array}
  \ast
  \begin{array}{c|c}\omega_2&H_2\\\hline T_2&A_2\end{array}
  \underset{\beta}{=}
  \begin{array}{c|cc}
    \omega_1\omega_2 & H_1 & H_2 \\
    \hline
    T_1 & A_1 & 0 \\
    T_2 & 0 & A_2
  \end{array}
$
\hfill $\displaystyle
  \begin{CD}
    \begin{array}{c|c}
      \omega & H \\
      \hline
      u & \alpha \\
      v & \beta \\
      T & \gamma
    \end{array}
    @>tm^{uv}_w>\beta>
    \begin{array}{c|c}
      \omega & H \\
      \hline
      w & (\alpha+\beta)\sslash({u,v\atop\to w}) \\
      T & \gamma
    \end{array}
  \end{CD}
$
\hfill $\displaystyle
  R^\pm_{ux} \underset{\beta}{=}
  \begin{array}{c|cc}
    1 & x \\
    \hline
    u & t_u^{\pm 1}-1
  \end{array}
$

\vskip 1mm $\displaystyle
  \begin{CD}
    \begin{array}{c|ccc}
      \omega & x & y & H \\
      \hline
      T & \alpha & \beta & \gamma
    \end{array}
    @>{hm^{xy}_z}>\beta>
    \begin{array}{c|cc}
      \omega & z & H \\
      \hline
      T & \alpha+\beta+\langle\alpha\rangle\beta & \gamma
    \end{array}
  \end{CD}
$
\hfill $\displaystyle
  \begin{CD}
    \begin{array}{c|cc}
      \omega & x & H \\
      \hline
      u & \alpha & \beta \\
      T & \gamma & \delta
    \end{array}
    @>{sw^{ux}_{th}}>\beta>
    \begin{array}{c|cc}
      \omega\epsilon & x & H \\
      \hline
      u & \alpha(1+\langle\gamma\rangle/\epsilon)
        & \beta(1+\langle\gamma\rangle/\epsilon) \\
      T & \gamma/\epsilon & \delta-\gamma\beta/\epsilon
    \end{array}
  \end{CD}
$

\vskip 1mm Constraints. $\bullet$ Column sums are monomials minus 1.

\rule{\textwidth}{1pt}

$\beta$-better calculus:

\vskip 1mm  $\displaystyle
  \begin{array}{c|c}\omega_1&H_1\\\hline T_1&A_1\\ - & \sigma_1\end{array}
  \ast
  \begin{array}{c|c}\omega_2&H_2\\\hline T_2&A_2\\ - & \sigma_2\end{array}
  \underset{\beta_b}{=}
  \begin{array}{c|cc}
    \omega_1\omega_2 & H_1 & H_2 \\
    \hline
    T_1 & \omega_2A_1 & 0 \\
    T_2 & 0 & \omega_1A_2 \\
    - & \sigma_1 & \sigma_2
  \end{array}
$
\hfill $\displaystyle
  \begin{CD}
    \begin{array}{c|c}
      \omega & H \\
      \hline
      u & \alpha \\
      v & \beta \\
      T & \gamma \\
      - & \sigma
    \end{array}
    @>tm^{uv}_w>\beta_b>
    \begin{array}{c|c}
      \omega & H \\
      \hline
      w & (\alpha+\beta)\sslash({u,v\atop\to w}) \\
      T & \gamma \\
      - & \sigma
    \end{array}
  \end{CD}
$
\hfill $\displaystyle R^\pm_{ux}
  \underset{\beta_b}{=}
  \begin{array}{c|cc}
    1 & x \\
    \hline
    u & t_u^{\pm 1}-1 \\
    - & t_u^{\pm 1}
  \end{array}
$

\vskip 1mm $\displaystyle
  \begin{CD}
    \begin{array}{c|ccc}
      \omega & x & y & H \\
      \hline
      T & \alpha & \beta & \gamma \\
      - & \sigma_x & \sigma_y & \sigma
    \end{array}
    @>{hm^{xy}_z}>\beta_b>
    \begin{array}{c|cc}
      \omega & z & H \\
      \hline
      T & \alpha+\sigma_x\beta & \gamma \\
      - & \sigma_x\sigma_y & \sigma
    \end{array}
  \end{CD}
$
\hfill $\displaystyle
  \begin{CD}
    \begin{array}{c|cc}
      \omega & x & H \\
      \hline
      u & \alpha & \beta \\
      T & \gamma & \delta \\
      - & \sigma_x & \sigma
    \end{array}
    @>{sw^{ux}_{th}}>\beta_b>
    \begin{array}{c|cc}
      \omega+\alpha & x & H \\
      \hline
      u & \sigma_x\alpha & \sigma_x\beta \\
      T & \gamma & \delta+\frac{\alpha\delta-\gamma\beta}{\omega}
        = \frac{(\omega+\alpha)\delta-\gamma\beta}{\omega} \\
      - & \sigma_x & \sigma
    \end{array}
  \end{CD}
$

\vskip 1mm Constraints. $\bullet$ Sum of column $x$ is $(\sigma_x-1)w$.
\quad$\bullet$ Likely, $\omega^{k-1}\mid\Lambda^kA$.

\rule{\textwidth}{1pt}

From \href{http://drorbn.net/AcademicPensieve/2012-05/}{2012-05}/A Higher Minors Experiment:

\includegraphics[width=7in]{HigherMinors.png}

\vfill
\rule{\textwidth}{1pt}

{\bf To do.}
$\bullet$ Consider a verification program.
\quad$\bullet$ Add $dm$ formulas.
\quad\colorbox{yellow}{$\bullet$ Add Burau calculus.}
\quad$\bullet$ Add the conjugation relation.
\quad$\bullet$ Add the MVA formula

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\endinput